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```XIV. SATURATION OF NUCLEAR FORCES

The binding energy and volume of nuclei are proportional to A.

the mass number. This is not in accord a villi a, law of force which
gives equal interactions between all pairs of particles in the nucleus
for there are then A. [A — l)/2 distinct interacting pairs and a
binding energy at least proportional to A (A — l)/2 might be
expected, if not to a higher power of A. due to increased packing
with more interaction. Instead, the nuclear binding energies
seem similar to the internal energies of bulk matter, in which 2
pounds has twice as much energy and volume as 1 pound.

To account for this phenomenon of "saturation of nuclear
forces," in which one particle apparently interacts with only a
limited number of others, various hypotheses have been made,
and various other assumptions about the nature of the forces can
be shown to bo impossible.

Among the impossible assumptions is that which has been used
in this book so far, namely, an ordinary potential independent of
the angular momentum, because it is easily shown that such a
potential does not give saturation. This is so even if the Coulomb
repulsion of the protons is taken into account. The proof can be
carried out with various degrees of exactness, using the variational
method. This method is based ou the Schrodinger variational
theorem which states that the quantity

9. = \$f1B\$ dr/jy dr (1-14)

is a minimum when ^ is the correct eigenf unction of the lowest
eigenvalue E (i of //, and the minimum value of Q is E\$. Thus, it
the assumed Hamil Ionian operator representing the interaction
of the particles in a given nucleus is sandwiched between any
arbitrary ip in the expression for 11, the value of must be greater
(i.e., less negative) than the correct energy of that nucleus. The
simplest i^'s are plane waves inside a box representing the nucleus.
If the size of the box is adjusted to give as low an as possible,
this size comes out about equal to the range of nuclear forces,
which is clearly much too small. Further, it gives a potcntia

80

SATURATION OF NUCLEAR FORCES

SI

nergy proportional to A 2 , and a kinetic energy proportional to
a# The size of the coefficients of these powers is such that the
J otential energy dominates for A > 50 ; for .4 = 238 the binding
energy is greater than 238 mass units. This is convincing evidence;
that the ordinary potential will not work, and this is true inde-
pendently of the shape of the potential (square well, exponential,
Gaussian, etc.).

What is needed is a potential which prevents the particles from
getting too close together. A. poten fial repulsive at short distances,
originally used by Morse for molecules, has been explored by
Schiff and Fisk; the only objection is that the high repulsive
potential may give relativistic difficulties if it gets above
2Mc 2 ~ 1800 Mev, for a proton in such a state would have nega-
tive kinetic energy. However, the idea of a repulsive potential
has not been followed up sufficiently.

EXCHANGE FORCES

In the first paper on nuclear forces, TTeisenberg proposed, in
order to explain the saturation of nuclear forces, that these forces
are "exchanger" forces, similar to the force that binds ordinary
chemical molecules. Without inquiring into the origin of these
exchange forces, let us write down the various types of exchange
forces that can exist between two particles, and then examine the
effects of these forces on the properties of the deuteron, and on the
saturation of the binding energy.

For an ordinary (non-exchange) central force the Schrodinger
equation for two particles is (in the center-of-mass system) :

[(n 2 /M)V 3 + Etyixu % frj, ffjO

7(^(^*3* <rt„vz} (W5)
Wigner

In nuclear physics, such forces are called Wigner forces. The
interaction does not cause any exchange between, coordinates of
the two particles. Another type of interaction is bile that inter-
changes the space coordinates of the two particles in addition to
multiplication of \$ by some V(r); for such an interaction, the
Schrodinger equation is :

[{ii 2 /M)V 2 + Emt h t,, ffu ad = FWtffct, r,, <r u *a) (146)

Major ana

XIV. SATURATION OF NUCLEAR FORCES

The binding energy and volume of nuclei are proportional to A,
the mass number. This is not in accord with a law of force which
gives equal interactions between all pairs of particles in the nucleus,
for there are then A (A — J.)/2 distinct interacting pairs and a
binding energy at least proportional to A(A — L)/2 might be
expected, if not to a higher power of .4 clue to increased packing
with more interaction. Instead, the nuclear binding energies
seem similar to the internal energies of bulk matter, in which 2
pounds has twice as much energy and volume as 1 pound.

To account for this phenomenon of ''saturation of nuclear
forces," in which one particle apparently interacts wdth only a
limited number of others, various hypotheses have been made,
and various other assumptions about the nature of the forces can
be shown to be impossible.

Among the impossible assumptions is that which has been used
in this book so far, namely, an ordinary potential independent of
the angular momentum, because ft is easily shown that such a
potential does not give saturation. This is so even if the Coulomb
repulsion of the protons is taken into account. The proof can be
carried out "with various degrees of exactness, using the variational
method. This method is based on the Schro dinger variational
theorem which states that the quantity

9. = f\$£\$ dr/jf 2 dr

(1443

is a minimum when ^ is the correct eigenfunetion of the lowest
eigenvalue Eq of H, and the minimum value of V. is E\$, Thus, if
{hg assumed Ilamiltonian operator representing the interaction
of the particles in a given nucleus is sandwiched between any
arbitrary \$ in the expression for fi, the value of must be greater
(i.e., less negative) than the correct energy of that nucleus. The
simplest ^'s are plane waves inside a box representing the nucleus,
ff the size of the box is adjusted to give as low an 9. as possible,
this size comes out about equal to the range; of nuclear forces,
which is clearly much too small. Further, it gives a potential

80

SATURATION OF NUCLEAR FORCES

SI

energy proportional to A 2 , and a kinetic energy proportional to
A H , The size of the coefficients of these powers is such that the
potential energy dominates for A > 50; for A = 238 the binding
energy is greater than 23S mass mats, This is convincing evidence
that the ordinary potential will not work, and this is true inde-
pendently of the shape of the potential (square well, exponential,
Gaussian, etc.).

What is needed is a potential which prevents the particles from
getting too close together. A potential repulsive at short distances,
originally used by Morse for molecules, has bees explored by
Schiff and Fisk; the only objection is that the high repulsive
potential may give relativlstic difficulties if it gets above
2il/c 2 -^ 1800 Mev, for a proton in such a state would have nega-
tive kinetic energy. However, the idea of a repulsive potential
has not been followed up sufficiently.

EXCHANGE FORCES

In the first paper on nuclear forces, ITeisenberg proposed, in
order to explain the saturation of nuclear forces, that these forces
are ''exchange" forces, similar to the force that binds ordinary
chemical molecules. Without inquiring into the origin of these
exchange forces, let us write down the various types of exchange
forces that can exist between two particles, and then examine the
effects of these forces on the properties of the deuteron, and on the
saturation of the binding energy,

For an ordinary (non-exchange) central force the Schrodinger
equation for two particles is (in the center-of-mass system) :

[(h 2 /M)V 2 + Ety(r u r 2 , a u <r. 2 )

Wigner

In nuclear physics, such forces are called Wigner forces. The
interaction does not cause any exchange between coordinates of
the two particles. Another type of interaction is one that inter-
changes the space coordinates of the two particles in addition to
multiplication of # by some V(r); for such an in ferae tiou, the
Schrodinger equation is:

[(n 2 /M)V- + Ety(r h Z& « u <r 3 ) = V(r)f(i-2, *i, in,*& (146)

Major ana

S2

QUANTITATIVE THEORY OF NUCLEAR FORCES

Such a force is called a Majorana force. Two other possibilities
are: (1) the Bartlett force, with interchange of spin coordinates,
and (2) the Helsenberg force, with interchange of both space and
spin coordinates. The Schrodinger equations are respectively:

[(h 3 /'M)¥ 2 + E\^(r u t 2 , ft, a,) = 7(r)^(r 1; r 3 , *% ft) (147)

Bartlett

[(hVAf)V 2 + m(r h r 2 , ft, cj) = V(r)#(t2 t r,, <r 2 , ft) (14S)

ff&senherg

Effects of Exchange Forces. Exchange forces, with a V(r), are
central forces and do not cause mixing of I's. However, if a tensor
force is used instead of V(r) as the multiplying potential, I's are
mixed and the quadrapole moment of the deuieron may be ex-
plained as before. It should be pointed out that the tensor force
does not by itself lead to saturation; this was proved by Volkolf
(Phys. Rev. 62, 134).

Majorana Force. The Majorana interaction replaces (r) by
(— r) in •■]/-. Using the well-known behavior of the wave function
on such an inversion, the Schrodinger equation (146) may be
rewritten

UhVAOV 2 + EU® = (-l) ! 7(r)^(r) (149)

This is equivalent to having an ordinary potential that changes
sign according' to whether I is even or odd, and is independent of
spin. Since the experimental data discussed so far give informa-
tion on the potential only for 1= % we have as yet no direct evi-
dence as to wh.et.her the potential is "ordinary" or of the Majorana
type. Since the potential is attractive for I = 0, it would be
equally repulsive for I = 1 M the interaction, were totally of the
Majorana type.

Bartlett Force. Considering still a system of two particles, the
spin function is symmetric if the total spin B is 1, and antisym-
metric if the total spin is 0. Thus, the Schrodinger equation (147)
for the Bartlett force may be rewritten;;

[(h a /w 2 + mm = oo^'f-ww \$s®

This is equivalent to an ordinary potential which changes sign
between S = and £ = 1. Since we know from neutron-proton
scattering data that both the :i S and *M potentials are attractive,
the nuclear force cannot be totally of the Bartlett type.

SATURATION OF NUCLEAR FORCES

83

Heisenherg Force. Combining the arguments of the two last
paragraphs,' the Schrodinger equation (148) may be rewritten for
the Heisenbcrg force:

KhViiov 2 + Mum = c - iy +s+v mf(j) (m)

This is equivalent to an ordinary potential which changes sign
according to whether l+'\$ is even or odd. For example:, the
effective potential is:

for 3 5'

potential + V(r)

■v(f)

Sp

-Y(r)

+ 70')

(152)

The reversal of sign, between 3 \$- and ^-states indicates, as for
flu > Bartlett. force, that the nuclear force cannot he wholly of the
Heisenherg type. However, the difference between the S S and *£
neutron-proton well depths (about 21 and 12 Mev, respectively,
f or a _ 2.8 X I0~ ri cm) can be explained by assuming that the
interaction is about 25 per cent Ileisenberg or Bartlett and 75 per
cent Wigner or Majorana.

Exchange Forces and Saturation. The Bartlett spin-exchange
force does not lead to saturation of the binding energy per particle.
If the nuclear force were of the Bartlett type, heavy nuclei should
exist with all spins aligned where the number of interacting pairs
is A (A - l)/2, which leads to binding energy proportional to at
least the square of A.

However, the space exchange in the Majorana and the Ileisen-
berg forces does lead to saturation because of the alternation in
sign of the potential between odd and even I. For example, assume
the nuclear force is the Majorana type (we already know it cannot
bo more than about 25 per cent Heisenherg). Then saturation
should not be apparent in nuclei up to He 4 , for in He 4 the spatial
wave function can still be symmetrical in all four particles, without
violating the Pauli principle. We need only give antiparallel
spins (antisymmetric spin wave functions) to the two n.iutrons,
and likewise to the two protons. Thus the Majorana force does uot
alter the Wigner argument about the short range of the forces
based on the binding energies of He' 1 and lighter nuclei.

In the next heavier nucleus— I Ie B or hi 5 — the Pauli principle
can no longer be satisfied by spin wave functions alone; there-
fore, the spatial wave function must have at least one node. In

S-i

QUANTITATIVE THEORY OF NUCLEAR FORCES

other words, only four particles can be in an ,s-state, whereas the
last has to be put in a potato, and will therefore be repelled by
the other particles. He u and Li 5 should thus be unstable, in
agreement with experiment. This is a first sign of saturation.

To investigate saturation in heavy nuclei, one may use the
same variational method used at the beginning of the present-
chapter to prove that ordinary forces do not give saturation.
It i.s satisfactory that this calculation, in the case of the Majorana
force, does not lead to non-saturation. On the other hand, since
the variational method gives only a maximum to the true energy,
it cannot be used to prove that the Majorana force does give
saturation. Bat Wigncr ha.s given a conclusive argument that
saturation is achieved with the space-exchange Majorana force
(Proc. N T at. Acad. Set. 22, 662, .1936). The space-exchange part
of the Heisenberg force would also cause saturation.

SPIN AND ISOTOPIC SPIN

It is often convenient to write exchange forces in a slightly
different way . Sin c e f o i ■ t w o par tides

0"i ■ o% = +1 for S = 1

= -3 for S = 0, (153)

the Bartlett force between two particles can obviously be written as

[+v¥&s- i

-V(t), s =

W®(X + *i ■ <%)

(154)

The spin-exchange part of the Heisenberg force; could be written
in the same way.

In order to be able to use a similar notation for the space-
exchange part of forces, we introduce the concept of the charge
of a particle as a coordinate, i.e., neutron and proton are regarded
as different eigenstates of the same particle, called a nudeon. We
choose the symbol t for this charge coordinate and we define

My = }4 for the proton
M T m — ]/i for the neutron
T = Y % for both

(1-55)

SATURATION OF NUCLEAR FORCES

85

using + H in analogy with the spin coordinate. We also define;
the charge functions

Charge function = y for the proton
= 5 for the neutron

(156)

in analogy with the spin functions a and 8.

The nucleons must obey Fermi statistics in order to be con-
sistent- with the ordinary theory (this will become apparent
shortly, if it is not immediately obvious). Thus the total wave
function (including the charge function) for two or more particles

4> = #spafiB&3 'AspmW vWrKeO)

(157)

must be antisymmetric with respect to inter change of ail coordi-
nates of two iiueleons. We therefore look for symmetric and anti-
symmetric charge functions for two particles. Then; are four of
these, as given in Table 4.

TABLE 4

Twq-Pa:rticle Charge Functions

Rtspi'i.;- Net

State Functiou seating Symmetry Charge

I 7(l)v(2) He 2 symmetric 2e

II 5(I)5( 2 ) ™ 2 symmetric

III (1./v / 2)[t(1)5(2) +7(2)5(1)1 H- symmelEe e

IV (l/v / 2)[7(l)5(2) — 7(2)5(1)1 IT- antisymmetric c

Again, in analogy to spin, two quantum numbers are defined to
describe these functions: T to describe symmetry, and M T to
describe the net charge. These quantities have the values given
in Table 5.

TABLE 5

Quantum Nu-Mfstchs fok Chabge States

Stale

T

M r

I

1

I

II

1

-1

in

1

IV

W QUANTITATIVE THEORY OF NIT CLEAR FORCES

T is 1 for symmetric functions, for the antisymmetric function,
in analogy to spin. M r is the sum of the M r 's for the two nueleons.

In the literature t is called the "isotopic spin," T is called "the
total isotopic spin," and M T may be called the "component of r
in the direction of positive charge." T is analogous to total spin S,
and Mj to B z . Tor a given T, M T can have the values T, T — 1,
-■■,-T.

From Table 4 it is seen that a system containing two neutrons
or two protons has a symmetric charge function. Since we arc
assuming nueleons to obey Fermi statistics, the remainder of the
wave function (147) must be antisymmetric. This implies (cor-
rectly) Fermi statistics for neutrons and protons, disregarding
charge as a coordinate. But in a system containing a neutron and
a proton the charge function can be either symmetric or anti-
symmetric, and so also can the remainder of the wave function.
Therefore, the treatment of proton and neutron as two eigens fates
of the same particle does not in this case introduce any restric-
tions, consistent with the ordinary theory of statistics-.

it; is also convenient to introduce an operator t in analogy to
the cr operator, defined by its effect on the "charge coordinate"
M T . The eigenvalue of its absolute square is, again in analogy
with spin:

|t| 2 - Vf(T + 1) (158)

Then, just as for spin, in a system of two nucleons
Ti -To = +1 for T = 1
-3 for T =

(159)

Now the Heisenberg interaction can be written (letting V(r)
absorb the factor —1) as

¥>V{r){\ + T] -t.)

(160)

To prove this, we note that equation 100 changes sign according
to whether the charge part of the wave function (equation 157)
is symmetric or antisymmetric, i.e., according to whether the
product of space and spin functions is antisymmetric or symmetric,
which is just what is required according to equations 151 and 152.
The types of interaction between the two particles discussed so
far may now be summarized by listing the various types of oper-

SATURATTON OF NUCLEAR FORCES

87

ators, which when multiplied by some V{r) give the interactions
listed in Table 6.

Eisenbud and Wigner (Proc. Nat. Acad, Sei. 27, 281) have shown
that these interactions and their linear combinations are the onlv

TABLE G
Types of INTERACTIONS

Ordinary

Spin cxdiiinge

Spaee^spin exulian^o

Space exchange

Tensor

Tenser exchange

1

<H 0-2
Tl ■ T2

(<T1-0- 2 )(ti-to)

(<ri-r)(o-2-r)

(q-l ■ r) (0-2 ■ r) (Tl • Tn)

Ones possible under certain reasonable invariance requirements,
namely, excluding interactions depending on total eharge or on
the momentum. (The in tea 'action (tri H- ov>) • L depends on the
momentum.)

QUANTITATIVE THEORY OF EXCHANGE FORCES

In the last chapter, it was shown that the ground state of the
deuteron, the neutron-proton scattering, and the quadrupole
moment, of the deuteron could be obtained quantitatively by
assuming a neutron-proton interaction of the form

V

(even)

= -(1 - ¥& + US °T ' *z -I- ySrd.Kr) (161)

with

Jir) = V'o

r < a

J(r) =

r > a

g = 0.0715

V = 13.89 Mev

7 = 0.775

a = 2.80 X m~ 13 cm

— Rarita anil Sch winger.

The neutron-proton interaction (.161) applies only to states of
L = 0. The potential for other L is as yet arbitrary. If wc assume
in particular a fores of the type discussed in this chapter, i.e.,
depending only on the product of the isotopic spins X\ ■ r 2 , the
potential will depend only on the parity of the state. The poten-
tial for states of odd parity can only be determined from that for

88

QUANTITATIVE THEORY OF NUCLEAR FORCES

slates of even parity by making some assumption regarding the
exchange character (or dependence on Tj ■ fa) of the forces.
Rarita and Sch winger chose; to investigate three potentials which
were suggested by three types of meson theory (see Chapter XV) :
I. Symmetric meson theory.
II. Exchange forces, or charged meson theory.
III. Ordinary forces, or neutral meson theory.
These potentials are:

I.

II.

III.

" =

— }ifi • t 2 o-i

0"2 V&yim

[■' =

(-1)'

v

* even

V =

V e veil

(162)

where V cvm is given in equation 161.

For ordinary forces III, the potential in odd states is the same
as for even. Exchange forces II, on the other hand, have opposite

sign in odd states. To determine the behavior of the force sug-

TABLF 7

PriOFEitTiES ov a Neutron -Pkotcn System

Is o topic

Slate

Parity

Spin

s

Spin T

<TJ ■ CT2

T1-T2

>iS

oven

1

-3

1

3 S

even

1

+1

-3

!p

odd

-3

-3

3p

oild

1

1

-: 1

+1

gesled by the symmetric theory I, Table 7 of values of crj ■ ffs
and ti ■ t 2 has been construe ted for even and odd states of both
the singlet and the triplet types.

From equation 162 and Table 7, the symmetric theory (I) gives:

"f''odd = ~~ /a f'evcii , (M ,

(163)

Todd — ~~ 3 Y even

The three types of forces may now be compared with experi-
ments by computing neutron-proton scattering at high energy.
The energy chosen by Rarita and Sell winger was 15.3 Mev, for
which P-wavc scattering begins to be important. The P-wave
scattering is to be computed with the aid of equations 162 and 163,

SATURATION OF NUCLEAR FORCES 89

which give the potentials acting in the P-state. It should be noted
that in contrast to the usual theory in which a single phase shift 5 1
is computed for scattering in the P-state, three phase shifts ij ,
i} U and 5j 2 must be computed for scattering by the 3 P -, 3 Pi-, and
3 P 2 -states, respectively. The reason for this is that the effective
potential well for each of these three states differs because of the
presence of the non-central tensor force S&, Tn fact, the operator
S 12 has definite values (-4 and 2) for the states ;J P and S P X

TABLE 3

Well Depths in the Netjtbal Theory

Effective Well Depth with
Stale "Ordinary Forces"

3 P 29.2 Mcv (repulsive)

3 P i - 3 5 . 4 Me v (attracti ve )

3 P 2 - 9.6 Mev (attractive)

which occur unmixed and must therefore be eigenf unctions of Si 2-
The 3 P 2 -state has a fairly definite value of S v > (—%}, since at
15.3 Mev it is only slightly coupled to the 3 P 2 -state. (See Chapter
XIII for a discussion of how 81\$ couples states of different L but
the same J.)

Rarita and Schwinger (Phys. Rev. 59,556, 1941), using equation
161 and the values of S VA just quoted, give the effective well depths
for the 3 P-statcs in the neutral theory III as shown in Table 8.

TABLE 9

Phase Shifts in 3 Po, 3 Pt, and 3 F 2 States

Tlieoi-y

11!

VI

12

I

0.074

-0.05-1

-0.017

II

0.531

-0.114

-0.046

III

-0.102

0.995

0.073

The potentials of the charged theory II have opposite sign to the
tabulated values; those of the symmetric theory I have: opposite
sign and a, re one-third as large. (See equations 1(32, 163.) The
phase shifts for each of the three theories, using these well depths,
are given in Table 9.

Note that the phase shifts in Table 9 for theory I are small be-
cause potentials arc used which are only one-third as large as for the-

90

QUANTITATIVE THEORY OF NUCLEAR FORCES

pries II and III. ( S e e eq uati on 1 G3 . ) N o to al so that th e s igns o f the
phase shifts art; opposite in theories II and III because this is
also true of their potentials. (See equation 162.) Note further that
really large phase shifts occur only for strong attractive potentials,
i.e., tjo in theory 11 and 17! in theory III.

If the scattering contributions from the 3 P-states are added up
with the proper statistical weight (2,1 + 1) the total scattering
for 3 JP-states at 15.3 Mev is found to be:

1.

II.

III.

<r(8) - X 2 (0.0038 + 0.0045 cos 3 0)
<*{#) - X- (0.103 - 0.002 cos 2 8)
<r(8) = X 2 (0,187 + 0.687 cos 2 0)

(164)

with 4ttX 2 = 0.082 X 10 -34 cm 3 .

The scattering is also computed for the (?Bi + ? 'Bi) state. This
is added to equation 164, taking proper account of interference
terms with the result that the total triplet scattering in barns
becomes :

I. v(8) = 0.080 (0.983 + 0.002 cos fl + 0,051 cos 2 3)

11. a{0) = 0.746 (0.98G + 0.193 cos 3 + 0.041 cos 2 (?)

Ill <f(g) = 1.165 (0.857 + 0.849 cos + 0.429 cos 2 0)

(165)

The quantities in equation 165 are so normalized that the numbers
outside the parentheses represent the total cross sections.

A corresponding calculation for the l P and i S scattering gives:

1. a(0) = 0.444 (0.939 - 0.438 cos 3 + 0.182 cos 2 6)

IT. &(§) = 0.424 (0.985 - 0.240 cos 8 + 0.044 cos 3 0)

111. a(0) = 0.437 (0.955 + 0.498 cos + 0.1.34 cos 2 8)

where the potentials used in the 1 P state were :

I. 'V( l P) = -37( 1 .8) = 4-35.7 Mev"

II. Fit 1 ?) = -Vi'S) = +11.9 Mev

TIL V ( ' P) = V(\S) = -U.9 Mev

(166)

(107)

Note that the difference between a repulsive force (I and II) and
an attractive force III is shown by the sign of the teiTn in cos 9 in

SATURATION OF NUCLEAR FORCES

01

I

0.621 barn

II

0. 6G6 barn

III

. 983 barn

equation 106, which represents interference between the l P and
the l S states.

The total cross section can be obtained by adding the triplet
and the singlet scattering in a 3-to-l ratio. The three theories
give hi fractions of a barn the values shown in Table 10.

The total cross section should not be used by itself to mate a
definite decision between the three theories since it is influenced

TABLE 10

Theoretical Xijutr on' -Proton ScATTEHora at 15.3 Mev

Theory Total Cross Section Angular Distribution

1 - 0.080 cos + 0.077 coir 8
1 + . 126 cos -f . 042 cos ! 8
1 -h . 032 cos + . 4r>7 cos 2 8

by the range and the shape chosen for the interaction potential.
On the other hand, the angular distribution is good evidence for
the existence or non-existence of strong P-scattering, and also
gives the sign of that scattering— thus providing direct informa-
tion about the exchange nature of the neutron-proton force.

For comparison with experiment, we may note from Table 10
that at 15.3 Mev, theory I gives a weak backward maximum,
theory II a weak forward maximum, and theory III a strong
forward maximum.

EXPERIMENTS ON NEUTRON-PROTON SCATTERING

Total cross sections can bo obtained by measuring the absorption
of neutrons in paraffin and correcting for the presence of carbon.
Angular distributions have been measured by Amaldi and others
(N atu rwissen seh af ten 30, 582, 1942; also Ricerca scientifica
1942), using the recoil protons projected from a paraffin foil.
The proton directions are determined by the use of a coincidence-
counter "telescope." Proton ranges, hence energies, are deter-
mined by the simultaneous use of absorbing foils.

In the center- of-m ass system, conservation of momentum re-
quires that the neutron and proton leave each other in opposite
directions — i.e., at angles 8 and 180° — 8 to the incident neutron,
respectively. In the laboratory system, the two particles leave
at right angles to each other, and the angle between proton and
incident neutron is 90° — 8/2.

<Xi

QUANTITATIVE THEORY OF NUCLEAR FORCES

Amaldi found that the number of protons projected forward
was small, corresponding to weak neutron scattering in the back-
ward direction, 6 = ISO 3 . This is in agreement with ordinary
forces III and in contradiction to exchange and symmetric theories
II and I. Amaldi measured R = o-(180°)/V(90 D ), the angles
being the neutron scattering in the center-of-mass system. His
results are given in Table 11 together with their quoted accuracy.

TABLE 11

High-Energy Xeutu m -Proton SciSTBKnJS (Amm.iit)
It

E (in Mev)
12.5
13.3

14.0

= aCLScryM'JO )

0.71 ± 0.04
0.53 ±0.03
0.52 ±0.03

The values of 7? at 15.3 Mev computed from the cross-section
formulas in Table 10 give for the three theories:

I. R - 1.157

II. R = 8.916

III. B = 0.525 (168)

On the other hand. Champion and Powell (extension of experi-
ments reported in Proe. Hoy. Soe. 183, 64, 1944). using neutrons
of similar energy and using photographic techniques, find that the

scattering is practically isotropic. However, their experimental
data have less good statistics and greater correction factors than

Amaldi's.*

More definite evidence contradicting Amaldi's results comes from
measurements of the proton-proton scattering at energies of 14.5
Mev by It. R. Wilson and collaborators (Phys. Rev. 1947). Al-
though these experiments are preliminary, they indicate a slight
repulsion in the P-state. They might be reconcilable with ex-
change forces or with zero forces in the /'-state, but they appear
to fit best to a force of the cr, -0-2 type and they certainly contradict
an ordinary force such as would be required by Amaldi's experi-
ments. There is, of course, the logical possibility that neutron-
proton and proton-proton scattering are different, but. in any ease
the present state of this subject is inconclusive and more accurate
measurements are urgently needed.

* Laughlin .and Kruger (Phys. Rev. 71, 736, 1947) also tind isotropic dis-
tribution (at 12-13 Mev).— Note added in pro>>f.

SATURATION OF NUCLEAR FORCES

03

If Amaldi's results are correct they imply that the forces in the
P-state art; attractive, and they support the theory of ordinary
forces III. Unfortunately, this result cannot be easily reconciled
with the saturation property of nuclear forces.*

* Experiments carried out with, the 184-irich. cyclotron of the University of
California at the end of 1946 demonstrate definitely the exchange nature of
the forces between neutron and proton. It was shown in these experiments
that a neutron of about 100 Mev will produce protons mostly in the forward
direction and with energies neatly equal to J 00 Mev. This had been predicted
by Wick for high energy collisions between neutrons and protons. If the
forces were ordinary forces the proton would in general receive an energy
of the order of the depth of the nuclear potential well, i.e., about 10 Mev. On
the other hand, if the interaction is of the exchange type, then neutron and
proton will change roles: the neutron will re Lain an energy of the order of
10 Mev and the proton will take almost the entire energy. When this note
was written it had not been established whether the forces are of the pure
exchange type or of the type corresponding to the symmetrical meson theory.—

XV. SKETCH OF THE MESON THEORY
OF NUCLEAR FORCES

This theory is presented although it Iras so far not given any
results in quantitative agreement with empirical facts on nuclear
forces. However, it may give a valuable point of view.

The Coulomb force between two charged particles can Ik; ex-
plained in terms of the interaction of these particles with the elec-
tromagnetic field. Similarly, the force acting between two nucleons
might fee described by a meson held surrounding the first particle
which acts on the second.

Moving charges produce a radiation field which can be quantized
and described in terms of photons. The "quanta'' surrounding
a nuclear particle arc called mesons. Yukawa, in initiating the
meson theory (Proe. Phy si co-Math. Soc. Japan 17, 48, 1935),
suggested that if the mesons are given a finite rest mass m. the
range of forces arising from the meson field will be h/mc, the
Compton wave length for the meson. If the range of nuclear
forces is assumed to be 2,8 X 10 _1B em, the meson rest mass
rest mass were discovered in cosmic rays two years later. In the
meantime, Erode- and Fretter have determined the rest mass to
be 202 ± 10 electron masses, giving a range of 2 X 10' I,! cm.

To determine the nature of the meson field and the correspond-
ing nuclear forces, an equation analogous to V- \$ = — 4irp must be
written for the static part of the electromagnetic field. A rela-
tivist! c equation suited for particles with no spin and a finite rest;
mass m is the Klein- Gordon equation :

?V + (l/nV)[(E - TO 2 - (mc 2 f]4t = -Ittp (169)

with

E = ih(d/di) (169a)

where p in this case is proportional to the density of nucleons. In
free space, V = 0. For a static meson field, according to equation
169a, we must put E = 0. Furthermore;, if there is one point-
nucleon at the origin, the Klein-Gordon equation, becomes

W% - {mefMH. = 4ir E / 1 5(r) (170)

94

MESON THEORY OF NUCLEAR FORCES

95

where o represents the Dirae ^function , and g L is a constant
replacing the electronic charge in electrodynamics.

The solution of this equation is

\$ = -(d/r) exp [-(jn.c/h)r\ (171)

and the potential acting on a second nucleon is given by:

V = tyfr 0"2)

where ffi and g 2 are the effective nucleonic "charges" or coupling
constants.

The Yukawa scalar meson theory just described produces the
required range for nuclear forces. Since in this theory the nuclear
particle does not change its nature (i.e., charge) we find that
according to tire theory the neutron-neutron, neutron -pro ton and
proton-proton forces are all equal. However, the theory does not
explain the spin-dependence of nuclear forces. Furthermore, the
forces are all "ordinary," whereas exchange; forces were found to
be necessary to explain the saturation of nuclear forces.

Since tire mesons discovered in cosmic rays were; all charged
either positively or negatively, a theory of charged mesons was
developed. According to this theory, the following reactions can
take place: p^# + A + m N ^ p + ^- (]73)

Thus protons and neutrons can transform into each other by the
emission or absorption of positive or negative mesons. The
interaction between two particles, 1 and 2. can take place, for
instauee, by the following scheme:

Pi -* Ai + ^ -Y 2 + M + - > i\ (174)

It is clear that such an interaction can only occur between a
proton and a neutron, not between two like particles. This is in
contradiction to experimental evidence and rules out the charged
meson theory, at least in tin; case of weak coupling between
nucleons and meson field (small value of g). Further, the charges
of particles J and 2 are exchanged in the process of emission and
reabsorption of the meson: therefore; this meson theory leads to a
force of the charge exchange; or Heisenberg type. This, while giv-
ing saturation, is in contradiction with experiment (Chapter XIV).
To explain the neutron-neutron and proton-proton forces which
are missing in the charged theory, a symmetric scalar meson theory
was developed, containing neutral, positive and negative mesons
described by three functions #i, fa, and ifo. To get spin-dependent

90

QUANTITATIVE THEORY OF NUCLEAR FORCES

V = g i

+ °V°"2

(175)

nuclear forces, the meson field must further depend on the spin of
the nucleon which generates the field. This is achieved by intro-
ducing into the Tlamiltonian of nucleon plus meson field, an inter-
action energy containing the factor cr ■ grad \$ where u is the
nucleon spin. In this case \$ must be a "pseudoscalar" since <r is
an axial and grad a polar vector. (A pseudosealar changes sign
-when the sign of the time is reversed, or on inversion of the Spatial
coordinates; under Lorentz transformations, it is invariant.)

Solution of the symmetric pseudosealar meson field equation
led to an interaction energy between two nucleons of the form

i r /3 :v A

-Tl-T 3 s ia i-.|-.-+-i
3 _ \r r r /

where i± = mr;/h.

The term in o-j - cr 2 provides the spin dependence of nuclear
forces, and the tensor force S l2 explains the existence and sign of
the quadrupole moment. All these features are in qualitative
agreement with experiment, as shown in the preeeding chapters.
Unfortunately, the high singularity of V at r = makes it impos-
sible to solve; the Schrodinger equation.

Two ways of saving the situation have been suggested: (1) to
cut off the interaction at some finite radius r u , i.e., to give the
neutrons and the protons a finite size, or (2) to mix two meson
theories in such a, way as to eliminate the undesirable singularity.

Tie assumption of finite sources (1) unfortunately caimot be
formulated in a relativistic invariant way. Furthermore, use of
the rigorous relativistic interaction between nucleon and meson
field leads to the reappearance of terms in \/r and l/r s in the
"mixed" theories, in higher approximations. Therefore there are at
present no trustworthy results of the meson theory of nuclear forces.

It should be noted that many of the statements made about the
spin and charge dependence of the nuclear forges have to be modi-
fied if the coupling between nucleon and meson field is strong, i.e.,
if many mesons are emitted simultaneously. The coupling con-
stant for an electromagnetic field is (?/lic = MV 7j "■ SB6&EI value,
whereas that for the meson field cf/hc ca 14 or % is considerably
larger. The divergence of the interaction at sma.ll distances
makes the interaction effectively even stronger. For this reason,
much effort has been spent to treat the strong coupling problem
in meson theory, but so far no results have been obtained which
throw light on the problem of nuclear forces.

C TOPICS NOT RELATED TO NUCLEAR
FORCES

XVI. BETA DISINTEGRATION

In Chapter VI, experimental evidence was given for the hy-
pothesis of the production of neutrinos of rest mass and spin }/ 2
in 0-decay processes. This assumption made possible the conserva-
tion of energv and spin. The first detailed theory of the process
was given by Fermi (Zeitschrift fur Fhysik 88, 161, 1934). A
modification which seemed necessary but was later abandoned
was the work of Konopinski and Uhlenbeck (Pbys. Rev. 48, 7,
1935). A summary is given by .Konopinski (Rev. Modern Phys.
15,209, 194.3).

Fermi introduced a new interaction between the nucleon and
the two light particles, electron and neutrino. His interaction
was chosen in analogy with the interaction between charges and
electromagnetic field in quantum electrodynamics. (This analogy
was also used in the last chapter in connection with the meson
theory of nuclear forces.) The heavy particles are to act as sources
and sinks of the light particles.

If the Hamil Ionian of the interaction between the proton,
neutron, and electron -neutrino fields is H ; then the number of
transition processes per unit time is

(2^/h}|J> fitI .*^,,^ 3 • pQB)

(176)

where p{S). — the number of final states of the system per unit
energy interval
=s initial state of the system
= u^. = initial state of the nucleon.
= Mfin. ' &fee. ' <?n. = Spal state of the system
= (final state of nucleon) ■ (final state of electron)
■ (final state of neutrino) .

to

Fermi's assumption for // was essentially

Jfe* Hto„ (It - gfu m * fee* <^.ifin. & (177)

(neglecting relativistic corrections which are important only if the
heavy particle has high velocity) where fee. arul S8t. arc to bt;

97

98 TOPICS NOT RELATED TO NUCLEAR FORCES

evaluated at the position of the nucleoli, and therefore the integral
is over the coordinates of the nucleoli alone. This is similar to
the case of electrons and light: a charge can only interact with a
light quantum, when they are at the same place. The constant §
which determines the strength of the interaction must be found
from experiment. It has the dimensions erg ■ cm 3 , since 1^1^.
and <p n . are Lo he normalized per unit volume.

Note that we use ^i^,*, bat p a , (without a star). This corre-
sponds to the emission of an election but the absorption of a
neutrino. However, this absorbed neutrino can be taken from a
state of negative energy which corresponds to the emission of an
'■'antineutrino." Owing to the absence of charge and magnetic
moment, an antineutrino is equivalent to a neutrino. The formu-
lation (177) is therefore equivalent to the emission of an electron
and a neutrino, and it is a mathematical convenience to have
formally one particle absorbed and one created. The positron
emission would be described by #ejes.*k*

Since the neutrino has very little interaction with anything, its
wave function may be taken as a piano wave. If p n . is the; mo-
mentum of the emitted antinouirino, then — p n . is that of the
absorbed neutrino of negative energy, and

Pa. = T ! -exp (-i p ru - r/fi)

(178)

where V is the volume of a box in which the wave function is
normalized. The factor V~ 1 ' 2 may be omitted if a unit volume is
used for the normalization. \$&«. should be a Coulomb wave func-
tion; but if Z the charge number is small, the Coulomb energy of
the electron can be neglected in comparison with its kinetic energy
and. a plane wave can be used for the electron wave function.
The number of final states per unit energy is

P (m =

(Volume element of mo- (Volume element of mo-
mentum space of electron) men turn space of neutrino)
(Volume of phase space per (Volume of phase space per
electron energy state) " neutrino energy state) XdE

= (Jeta 2 #ete \$»ri«J IP". 3 # B . doi n )/(2rhf d.E n ,
where do> elvi .da) n _ are elements of Solid angle.

(179)

BETA DISINTEGRATION

The result for the transition probability of an electron into
dS d[iC . and solid single i&f&w (integration over all directions of the
neutrino has been carried out) is

G 2 mc? I r
§^~h~P^

exp

-i(pn.+P C ]o .)-

2 , m

rfre( e 3 -l)^ U -0 2 rfe-
(180)

with G = (f//mc 2 )(h/mc) % e = E^Jmtr, V r - I. = p^/jrac,
eg = E^, llihM Jmc 2 . A plane wave has been substituted for the
electron wave function.

Just as in the theory of atomic transitions, there will be selection
rules for /3-deeay processes. If p„ h: ... and p n _ are both of the order
of magnitude mc. as is usually the case, the exponent (p„ -f- p (!let .,)
■ r/h will be of the order of magnitude;

4 X 10"" cm

h/mc 3.85 X I0~ n cm

1

100

(181)

(R = nuclear radius; medium-weight nuclei have been chosen.)
Thus, exp [i(p n . + p ek .c.) ■ r/h] will be nearly 1, and the matrix
element in equation 180 reduces to M = J&fia.**^. dr ; i.e., to an
expression depending only on the state of the nucleon before and
after the transition. M is determined by the nuclear wave func-
tions. In parti cular, the orthogonality of the nuclear wave func-
tions for states of different angular momentum I gives the selection

ni,e: M fi implies AT = (182)

Such transitions are called allowed. Transitions for which M =
are called forbidden; in this case the exponential in equation 180
must be expanded in a power series; the order of the forbidden
transition is the number of the first term in this power series which
gives a n on -vanishing result for the matrix element. Because of
the estimate (181), the probabilities should decrease by a factor
of about 10 4 with each order.

ALLOWED TRANSITIONS

The only depend ence of the allowed transition probability on
the electron energy is through the volume clement in momentum
space. The energy spectrum of electrons is therefore

KB de ~ el/e s - Ifo - tf d* (183)

100

TOPICS NOT BELATED TO NUCLEAR FORCES

Since eo is unknown, the experiments have to yield a value of t#,
while giving a check on the theoretical spectrum.. This is easily
done by making a "Kurie plot." In this plot, the quantity

F(e) = V;V(f)/£(i 2 - 1) H

(184)

According to
hould vie Id a

F<t)

(as observed) is plotted against the energy &

equation 183, F(e) <--■ e () — e; therefore the plot

straight line which cuts the e-axis at eg.

The only nucleus which checks this proportionality exactly is

In 114 , measured by Lawson and Cork (Phys. Rev. 57, 982, 19-10).

Hero to = 1.99 M.ev (which is
high enough to make the ex-
periments on the /3-rays fairly
easy) and the lifetime is 72
seconds. Luckily this short-
transition; see Chapter IV).

There are experimental dif-
ficulties in flic measurement
of the energy spec tin of most

Thick
target

0.615 Mev E

Fig. 13. Kurie plot of the positron
spectrum from Cu .

other /^radioactive nuclei which result from either the low energy
of the electrons or the short lifetimes. Cu M measured by A. W.
Tyler (Phys. Rev. 56, 125, 1.939) emits both positrons and elec-
trons. The positron spectrum was measured both for thick target
and thin target (thick and thin relative to the electron range).
The Kurie plots are shown in Pig. 13. It is not known whether
the portion AB of the thin target curve is spurious or results
from another decay process (to an excited state of JMi* 54 ) with a
very low energy limit.

The thick target curve is typical of the experimental evidence
which lead Konopinski and L'hlenbeck to introduce then alterna-
tive theory (Phys. Rev. 48, 7, .1935). They proposed using the
time derivative of the neutrino wave function &&J&i in the transi-
tion probability instead of ip. Since &/>/ dl o-< (sq — e)^ this led to

spectrum

eVe 2 - l(«u - <0 4 de

A 7 k-u de>~^ e - *

(185)

thereby moving the maximum of the spectrum to lower electron

BETA DISINTEGRATION

101

energies. To make a Kurie plot of this, the fourth root must be
used in equation 1S--1 instead of the second. Many of the experi-
mental data on thick targets then give straight lines but very high
values of €<>. Later experiments us ing thin targets showed that
the Kurie plots according to the Konopinski-TJhlenbeck theory
dropped off, as shown in Pig. 11, which demonstrated that the
straiglrt-line portion was accidental. Also, when the mass differ-
ences of nuclei became better known, the values of <e given by
the Konopinski-Uhlenbeck theory were shown to be much too
Mgh in all cases but that those given by the Fermi theory agreed
with the measured mass difference.

A' 13 measured by Kikuchi et al. (Proc. Physi co-Math. Soc.
Japan, 21, 52, 1939); Lyman (Phys. Rev. 55, 1123, 1939); and

VV--V

Fir.. 11. Typical Kurie plot of the Konopinski- Ohlenbcck theory.

Townsend (Proc. Roy. Soc. A177, 357, 1941), is one case in which
the use of very thin targets still did not gore a Fermi distribution.
To account for such spectra it is usually assumed that several
decay processes an; taking place simultaneously, leading to various
energy levels of the residual nucleus. With N 13 — » C li + §
this is confirmed by the observation of a y-ra,y of about 280 kev
by Richardson (Phys. Rev. 55, 609, 1939). This y-ray is attributed
to the transition of the residual nucleus C 13 from its excited to
the ground state. Unfortunately, various experimenters disagree
on the relative intensities of the 7-rays and of the two compo-
nents of the /3-spectrum, and on the value of the upper limit of its
lower-energy component.

Coulomb Field. In expression 183 for the electron energy
spectrum no account has been taken of the Coulomb field. The
correct spectrum has a greater electron density at lo w ener gies.
There is no zero for e = 1 because the factor V« -l~s
(velocity) in the density of states Is canceled by a 1/v in the charge

■ I

102

TOPICS NOT BELATED TO XL" CLEAR FORCES

density of electrons at the nu ele us. The result i ng electron spectrum

is showa m Fig. 15.

Far positrons, fewer of low energy should be expected than the

number given by expression 183 because of the repulsion of the

positrons in the Coulomb field:
The Coulomb wave function of
the electron in expression 177
has a facto c cxp ( — 2-irZcr ,/hy) ,
which lowers the transition
probability considerably for low
velocities.

There are some disturbing
measurements by Backus (Rhys.
.Rev. 68, 59, 1915) on the ratio
., in the Cu fil ^-transitions:

Fig. 15. Energy distribution of
jS-mys with Coulomb field.

of positrons to electrons, N+/N.

Cu 64 -

f \m m + ,3-

[Zn^ + f

a calculable way at

N+/N"_ should be smallest and behave in
low energies; the experimental values were compared with the
theoretical prediction but the value of N_j_/N_ was found to be
ten times greater than predicted. These measurements should be
repeated. The disagreement can hardly be attributed to a failure
of 0-ray theory because the ratio of positron emission to /C-electron
capture was found to be in exact agreement with theory (Scherrer
et ah, Phys. Rev. 68, 57, 1.9:1:5}, and this ratio Involves parts of
the theory very similar to those in Backus' experiment,

The total transition probability, or reciprocal of the lifetime,
for /3-ray emission is found by integrating over the energy distribu-
tion (equation 180) to Ire

1/V b (G 2 /2tv*) • (mcVh)|ilir/<-(^)

(186)

G is a dimonsionless constant describing the strength of the inter-
action between electron-neutrino and the heavy particles: M is
the matrix element for the transition :

M = fu nn *(T)u in _(r)e- i( ^+*< ] -* /h dT

(187)

!

BETA DISINTEGRATION
F(e ) is the integral of the distribution in energy

f(& = £*W^ - *& - *f^

103

(188)

where e is the total energy available for neutrino and electron,
including rest mass, in units of the electron rest energy. W (*\$)
varies rapidly with e (1 , being approximately equal to ( ! /SO) e s for
e » 1 aurl to 0.216(fF — l) y ' for sg nearly unity. Thus r de-
creases rapidly with increasing e , but not as fast as in the case of
a-decay, where the transition probability is proportional to an
exponential of the energy. In Chapter II it was pointed out that
in natural a-deeay a factor of 2 in energy is equivalent to a factor

The matrix element M is in general not known because we have
very scant knowledge of nuclea.r wave functions. Even if we know
that the transition is allowed, we can in general say only that \M\
is between zero and one.

However, in some cases the value of M can be guessed to some-
what better than order of magnitude. For allowed transitions
(A/ = 0) , we have

M «/%*.%, 4r (189)

M will be near unity when the wave functions u fill . and u- m , are
nearly alike. Such is the ease for ,6*-transition between mirror
nuclei (Chapter II) (for which also the selection rule Al — is
likely to be fulfilled). Three examples of allowed transitions in
mirror nuclei are given in Table 12. The product iFfa) is remark -

TABLE 12

Allowed Transitions in Mihhob Nuclei

Reaction

t = half-life

eo

tFUn)

H 3 -a He 3 + (?~ + "

10* m

1.03

1400

C u -* B" +fT + v

1200 sec

2.S6

3500

So* 1 -» Ca 4! + D+ + v

. <J sec

10. OS

2500

Source: Konopinski, Rev. Modern Phys. 15, 209.

ably constant, confirming the theory underlying equation 186.
This constancy exists in spite of t varying by a. factor as large as
10 s . Furthermore, it, is reasonable that, tF is somewhat smaller
for the first situation than for the other two, for in a nucleus

104

TOPICS NOT RELATED TO NUCLEAR FORCES

containing only three particles we would expect u fm , and i%, to
be more nearly alike than in the heavier nuclei; so that | M \ would
be closer to unity in the light nucleus.

It is interesting to note that the Konopinski-TJhlenbeek theory
of /3-decay predicts variation by a factor of JO 5 between the prod-
ucts i-F for the various reactions in Table 12.

For nuclei of intermediate mass, the Coulomb repulsion already
introduces considerable asymmetry between the numbers of pro-

TABLE 13
Allowed Transitions in Intilkmediately Heavy Nuclei

Reaction
g?5 _> C1 3S + ?- + v
Cu fjl -> Zn M + \$~ + v

Cu M -> Ni i4 -T- 3+ + v
In 117 -h» Sn 1 *' 4

+ '

19,000
66,000

22,000
140,000

tons and neutrons (there are no more mirror nuclei), and pre-
sumably even greater' differences, between neutron and proton
wave functions in the nucleus. Thus, even for allowed transitions,
smaller matrix elements are expected for intermediately heavy
nuclei than for light, mirror nuclei. T his is borne put by the data
in Table 13.

In the heavy, naturally radioactive nuclei the matrix elements

TABLE 14
Allowed TsajtskSOKS in Naturally Radioactive Ndclf.i

Emitter

tF(ta)

RaB

50,000

UX S

270,000

are in general still smaller. This is borne out by the data in
Table 14.

Assuming \m\ ~ 1 for the lightest., mirror nuclei, can be cal-
culated from Ft. The result is

G « 10"

(190)

This corresponds to g ~ 10 ' ls erg • cni a . The smallness of this
coupling between electron-neutrino and the heavy particle is

BETA DISINTEGRATION

105

what makes (3-decay take place so slowly compared to other
nuclear reactions, except some a -radiation. It is safe to say that
ji-rayti are not emitted during nuclear collision*, but only at com-
paratively long times afterwards. For example, the lifetime of
protons in the sun due to the reaction

II + H -» D + r + v

(191)

is about 10 11 years, even with a density of about 100 and a tem-
perature of 2 X .10 T degrees C. (See Bethe and Critchfield, Phys.
Rev. 54, 248.) Even so, this reaction presents about the best
opportunity for /3-decay during a collision. The long lifetime of
the proton in the sun indicates an extremely low probability of
/'i-decay per collision.

The most fundamental (3-decay is that of the neutron

n —> II + j3 + v

(192)

The matrix element for this reaction should be exactly unity, as
the wave function for a single proton ought to be the same as that
of a single neutron. Measuring the lifetime of this reaction
should give an exa.ct value of G. However, this reaction is hard
to observe as the neutrons are removed much more rapidly by
other means (capture, diffusion) than by the above reaction.
Using the value of (J found above, the half-life for the reaction
(192) should be about 15 minutes. There is hope of making the
measurement with the large neutron fluxes now available in piles.

The second term in the Taylor expansion of the exponential
in the matrix element (1ST) will give a non-vanishing integral when
A/" = zhT. which transition was forbidden in the first approxima-
tion. Similarly, A/ = ±2 transitions become possible with the
third term in the expansion, and so on. For t = 2, the argument
of the exponential averages about 1/100 over the range of the
heavy par tick; wave function, so that \M{AI = =L.L)'| 2 might be
expected to be about JO" 1 times \M(&I = Q)| 3 . Actually, the true
wave function for an electron in the Coulomb field varies faster
than the plain; wave approximation used in equation 187, and the
factor 10" 1 becomes about 10~" for medium and heavy nuclei.
This correction does not help the higher forbidden transitions SO

106

TOPICS NOT RELATED TO NUCLEAR FORCES

much as the first. Higher ^ makes all forbidden transitions more
probable. Table 15 quotes experimental data from Konopinski
for forbidden transitions in light nuclei.

TABLE 15

IIaIjF-Ltves in FoRBIBOEN Transitions

Emitter t = Half-life m

First Forbidden, Transitions

&fa)

Li s

0.9 sco 24.5

2.S X 10 5

Ne 23

40 see 8

10 r '

Second Forbidden Transitions

pn

1.2 X 10 B see 4.37
Higher Forbidden Transitions

8.6 X I0 7

B<; m

KV^SCri 2.1

10 H

K «

5.10 16 sec 2.4

10 11

Source': Konopinski, Rev. Modem Phys. 15, 200.

GAMOW-TELLER SELECTION RULES

Tli ere is good evidence that the selection rule AT — for allowed
transitions is not generally adhered to. One example is the K-
eapture reaction ^ + K ^ L . 7 + p (m)

Li 7 is produced both in its ground state and in an excited state
about 440 kev above the ground state. The experimental ratio of
number of transitions to the ground state to number of transitions
to the excited state is about 1.0 to .1. This is about equal to the
calculated ratio, using equation 186 and assuming \M\ equal for
the two cases. From this and the absolute lifetime it may be
concluded that both transitions are allowed. However, we do
not expect both states of Li 7 to have the same value for I. The
best assumption is that the two states form a /'-doublet, with
I = Yi and I = % for excited and ground states, respectively.
Thus AT can certainly not be zero for both transitions.
Another example is the reaction:

He* -» Li 6 + /T + v (194)

Li a can be thought of as an ct-particle plus a deuteron. The
.a-particle has / = 0, and the deuteron has I =» 1. We expect,

BETA DISINTEGRATION

HIT

therefore, that Li a has I = 1, in agreement with experiment. In

the same picture. Tie 6 is an a-partiele plus two neutrons. In the
"ground state," the double neutron should have spin zero (ef.
Chapter XII), so that the same argument gives 7 = for He*.
An additional argument for this is that all nuclei containing even
numbers of neutrons and protons have zero spin as far as they have
been investigated. Thus AI = I, and the transition is forbidden.
But the experimental lifetime of the reaction shows that it is
"allowed," There are similar situations in the /3-decay of C i0 ,
F 18 , and Na 3U .

So it seems that there can be allowed transitions with A 7" = 1.

Gamow and Teller first allowed how this can come about. They
said that in considering possible interactions, one ought to include
all relativistically invariant combinations of the four wave func-
tions, u- m ., tifijj., &jfeM and 6,... For two wave functions, let us say
\f/ and 4>, there are five combinations which are co variant under
Lorentz transformations :

1. Scalar: ^* j3 <i> (Fermi theory).

2. Polar four vector, with components: ^* 6, f* a </>.

3. Tensor: \$*■&&■\$, V P a <t>-

4. Axial vector: \$* cr <*>, #* 7s *.

5. Pseudoscalar : \$* ,8 y 5 <j>.

where fi, a, and y 5 are Dirac operators and a is the usual spin
operator. (For details, see Konopinski's article.) To obtain a
relativisficaily invariant interaction, the corresponding combina-
tions of the wave functions of the light and of the heavy particles
must be multiplied; for example, the tensor combination of the
light particle wave functions with the tensor combination of the
wave functions u m , and i( fin . of the heavy particles. In this case
the Hainiltonian becomes:

V (tensor) = (f*j3<r4) ■ (u&^&TMia)

(195)

(The transition is still treated as though an antineutrino is
emitted.) Since the heavy particles are non-relativistic, the Dirac
operator ,3 for them is equivalent to unity; therefore, the net effect
of equation 195 is to place the operator cr between the heavy
particle wave functions fe. and « fln- . Therefore, the matrix
element for allowed transitions is now J ■'u-fi 11 .*<nii il , dr, and this
may be different from zero if the total sphi I changes by one unit,

mmm

10S

TOPICS NOT RELATED TO NUCLEAR FORCES

or by zero, in the transition. Thus M = ±1,0 can be "allowed"
for the tensor interaction.

The axial vector interaetion gives the same selection rule as the
tensor,

Al = 0, ±1 (106)

From the experimental data it seems that these Gamow-Teller
selection rules are correct. For instance, they explain the results
for He 6 , C 10 , F 18 , and Na~. However, the reaction

Be in -» B 10 + &~ + v (197)

differs from (194) only by the addition of an at- particle, so that
&I = 1 may again be expected for this reaction. But experiment
shows that this is forbidden. The same is true for the reaction

C 1

n m + £-- + v

(198)

which differs from (191) by two a-partieles. Thus the Gamow-
Teller selection rides, while;, explaining more than the Fermi rules,
still are in contradiction with many of fetes data.

K-capiure. The theory for A'-capture lias been worked out,
and is in good agreement with experiment. Scherrer el al. (Phys.
Rev. 68, 57) have measured the ratio of Z -capture processes to
positron-emission processes for Cd 107 (or loy ?), with the result:
320 =fc 20. The Fermi theory predicts 340. (The Konopinski-
Uhlenbeck theory gives 20,000, and is conclusively ruled out.)

XVII. THE COMPOUND NUCLEUS

In tins chapter, we are no longer concerned with the determina-
tion of fundamental nuclear forces, but with the more practical
problem of predicting cross sections for nuclear reactions, par-
ticularly those involving heavier nuclei the quantum states of
which are not known precisely. On the other hand, the presence
of many nuclear particles will make statistical methods practical,
and these are used in the theory of the compound nucleus.

The concept of the compound nucleus was initiated by Bohr m
1935. In order to get a clear picture of this concept, we shall
examine the difference between nuclear collisions and atomic

collisions.

For collisions between an. atom and a particle of high or moderate
energy, the Born approximation is valid because the incident
particle passes right through the atom practically undisturbed.
Slight deflections, inelastic collisions, and emission of radiation
are progressively less likely processes. The reason that particles
are likely to pass right through is that the atom is a loosely bound
structure. Another way of saying this is that the interaction of
atomic electrons with, say an incident electron of several thousand
volts, is much smaller than the incident energy- -which is precisely
the condition for validity of Bom's approximation.

Nuclear interactions, on the other hand, are of the order of 20
Mev, which is much greater than the kinetic energy of the incident
particle normally used, i.e., several Mev or less. This is precisely
the opposite of the conditions required for Born's approximation.
Here, the interaction energy is more important than the kinetic

energy.

Another difference: An electron striking an atom can be re-
garded as interacting with the average "Hartree" field of the atom.
This approximation is valid because the interaction with a single
electron is much smaller than the average interaction with all the
electrons. On the; other hand, the short range and the saturation
character of nuclear forces require that, nucleons interact only
with a small number of neighbors. Tims individual interactions

109

11.0

TOPICS NOT .RELATED TO "NUCLEAR FORCES

will be of the same order of importance as the average total inter-
action — and it will not be permissible to replace tire nucleus by an
average field.

The Bohr picture takes advantage of these large interactions
and describes them in terms of a compound nucleus. Tire theory
makes the following statements;

1. Any 'particle which kits the nucleus is caught. A new nucleus
is formed called the compound nucleus. The reason for this is
that an incident particle will interact with one or two nucleons,
transferring much of its energy to them and thus to the nucleus,
before penetrating it appreciably. Then it may no longer have
sufficient, kinetic energy to escape the attractive nuclear forces,
and is therefore caught.

2. The compound nucleus is long-lived compared to the natural

nuclear time. (This is the time for a neutron to cross the nucleus — ■

cm
say 10 ~ r2 cm/ 10" -' - =* 10 ~ 31 second.) The reason for this is
sec

that the compound nucleus, which is in air excited state (excitation

energy above the ground state = incident energy T binding

energy of oire particle), will live until this excitation energy, or a

reasonable fraction of it, is concentrated again on one particle.

3. The final break-up of the nucleus is independent of the mode
of formation, i.e., regardless of how- the nucleus was formed there
will be definite probabilities for decay into each of several possible
residual nuclei. This can be explained in terms of the long life-
time of the compound nucleus during which, complete statistical
equilibrium, is assumed to be established — thus the nucleus forgets
how it was formed; formation and disintegration can be regarded
as independent events.

For example, the ordinary Al nucleus ( l3 Al 2 ') can be formed as
a "compound nucleus" in a highly excited state from any of the

(199)

reactions ;

u Na 23 + W -» 1S A1 37 excited"

12 Mg 2E + Tl 2 -> 13 A1 27 excited

12 Xlg 2fi + H 1 -» "AF excited

"Al 27 + t -> IS AF excited.

The compound nucleus can then decay back, reversing the reac-
tion, into any of the nuclei just mentioned, or also into AF + %,

THE COMPOUND NUCLEUS

1.11

with a definite probability for each which is the same for all modes
of formation. The residual nuclei may also be left in excited
states, with probabilities which are also independent of the manner
of formation.

Formation of Compound Nucleus. The cross section for forma-
tion of the compound nucleus cy may be written in the form

H = tvR.% (200)

where H, is the nuclear radius, and £ is a useful parameter, called
the sticking probability, which is defined by this equation.

For fast nuclear particles, i.e., X «R{\ ~ 10" 12 cm for 200-kv
neutrons), the classical geometrical approach is valid since the
uncertainty in position of the particle is only X. The cross section
for capture of fast nuclear particles is certainly not greater than
vB 2 since the interaction is negligible ii the particle passes at a
distance from the nucleus. For slow neutrons, however, cross
sections greater than irR 2 are possible since the position of the
particle is poorly defined. To get a sticking probability which is
always <1, the definition is revised. We define the contribution
ffL to" the cross section due to particles of orbital momentum I,

andset *, - (21 + 1)*X% (201)

Then from general principles of quantum mechanics, £j must be
less than (or equal to) 1. Moreover, equation 201 reduces to
equation 200 for high energy since all values of I up to R/X will
contribute appreciably (cf ." Chapter IX, p. 38) ; £ is a weighted
average of % Neutrons were used in the above discussion to
avoid questions involving penetration of the potential barrier
winch would arise for protons and a- particles.

The Bohr statement, that, any particle which hits the nucleus
is eatight, is given more precisely by the equation

In other words, the sticking probability approaches 1 at, high
energies. This statement has been checked experimen tally with
high -energy neutrons especially by Amaldi and co- workers, by
Sherr, and by Graham and Seaborg. They find cross sections of
about it/ & irft 2 , with E given, by a formula similar to equation 3,
in good agreement with other methods of determining nuclear

112

TOPICS NOT RELATED TO NUCLEAR FORCES

Disintegration of Compound Nucleus. The probability that the
compound nucleus will disintegrate in a particular way is related
to the cross section for the corresponding inverse capture process
with some factors containing Ihe density of initial and final states.
This follows from considering a statistical equilibrium condition
between the compound nucleus and all the possible states of all
the residual nuclei into which it can disintegrate (similar to
Chapter XI, p. 60). In. equilibrium, the number of nuclei present
in a small energy range between E and E + dli will be proportional
to the density of states p(E) in that energy range, and to a Boltz-
mann factor. Since energy is conserved in the total system, the
Boltzniann factors cancel out and the condition for equilibrium
takes the form

PjtWx-fji « m&B^jk (203)

where p A and p Ti are the densities of initial and final states of the
system at corresponding energies, and the W's represent prob-
abilities for the direct and inverse processes.

For our process, A is the excited compound nucleus 'rath a
density of states pa(Ea) = I/Aaj where D is the average separa-
tion between neighboring states, at an energy E A above the ground
state of A- (Each state is counted according to its statistical
weight.) Wa-*b is &hs probability of disintegration of the com-
pound nucleus into a definite state of the residual nucleus B with
energy E\$ above its ground state, with the emission of a particle
(say neutron') of energy E. We^a is the probability that nucleus
B will capture this particle of energy E and produce a compound
state of excitation E A . Finally, ph(Ejj) gives the number of states
between E and E + dE available for tin: outgoing particle, viz.

47T-/J 2

PS

v(2*Ky s

(203a)

with p and v the momentum and velocity of the outgoing particle.
We now use the relation between the capture probability and the
capture cross section, which is

Wg^j

vc f (E)

(204)

for one neutron in a. box of unit volume moving with velocity
v => (2Efm) ' /2 , and the relation between the excitation energies E A

and Eg,

Ea

E - B

(205)

THE COMPOUND NUCLEUS

113

where E is the energy of the outgoing particle and B its binding
energy in the unexcited nucleus A.

Using all the relations just given, and setting I = in equation
201 (other I give veiy similar results), we now have a relation by
nieans of which the disintegration probability Wa~*8 — iVh can
be computed in terms of the sticking probability £g for the inverse
capture reaction :

n//;bi)(TVh) = m *\$fe ( 20e )

or, inserting 203« and simplifying:

T b /Da = fr/2* ( 206a )

This important equation relates the disintegration probability
Fb, leading to a definite state of the residual nucleus, to the level
spacing D A . For high energies, 6s approaches 1; for low energies
it is proportional to the velocity v of the emitted particle. Both
Da and T B can be deduced from experiment; D A and % B can also
be estimated from various statistical models for heavy nuclei
(Nuclear Physics B; Weisskopf, Phys. llcv. 52, 295, 1937; 57,
■172, 1940).

The disintegration probabilities tjj/fi >"'e also related to the
widths of the resonances observed in these reactions: since; the
total decay probability is

V/h

(i/h)^

B

the time dependence of the wave function is of the form

-iEl/h -lt/2h _ (J -i{li!-

f]'lf/h

(207)

(208)

(Note that the absolute square of the wave function gives the
occupation of the state and decays according to equation 207.)
Equation 208 has a Fourier transform * the absolute square of
which is:

(209)

(E' ~E) 2 + (r/2) £

Thus T has the same dimensions as E and gives the width at half-
maximum of the level, or resonance line. The quantity T B repre-

* Taking the Fourier t ns.n?f mid with respect to time of a- time-dependent
wave function gives the wave function 4i{B') in energy space.

__

114

TOPICS NOT RELATED TO NUCLEAR FORCES

sents a partial level width, i.e.. the contribution to V arising from
the disintegration into a definite end state B.

Since the compound nucleus must eventually decay, the cross
section for a reaction ending in state B is given by the cross section
for forming the compound nucleus, times Fg/F. Thus

(210)

and for fast particles :

ays = °7 iwr

ujb = rS'| Tg/T

(211)

Energy Distribution of Emitted Particles. From equation 206a
we see that V B is almost the same for any final state B, since the
sticking probability k B is a slowly varying function of the energy
of the outgoing particle. This information is useful in predicting
the energy distribution of the emitted particles. For example,
if we consider the inelastic scattering of neutrons

Z A + n->Z A -'^Z A ^n

(212)

and make use of the fact that the density of states in the residual
nucleus increases rapidly with excitation energy, then we see that
the residual nucleus will most likely be left in a fairly high excited
state and the emitted neutron, will come out with low energies.

The fact that emitted neutrons come out with greatly reduced
energies has been experimentally confirmed for many target nuclei.
Lead forms a notable exception to this rule. The reason for this
may be that the first excited state in this instance is quite high —
so that this rule would not be confirmed unless higher energy
incident neutrons are used. In fact, the incident energy must
be high enough so that the residual nucleus B possesses a great
many levels with an excitation energy less than the incident
kinetic energy E , in order that the statist! eal considerations used
may be valid.

Shadow Scattering. In neutron-scattering experiments a purely
wave-optical effect must be considered at high incident energies
(X « B). for which we have said the capture cross section is x.8 .
In this case, the nucleus can be regarded as a black sphere of
radius R which casts a shadow. This is described in the language
of wave optics by saying that just enough light is scattered in the

THE COMPOUND NUCLEUS

1 15

forward direction to cancel the incident beam. This would mean
a cross section for shadow scattering of irR 2 . Fur therm ore, to
cancel the incident beam behind the sphere, this shadow scattering
must be of the same energy, i.e., it represents elastic scattering.
According to an elementary wave-optical argument, the shadow
Scattering will be mostly confined to an angle X/fi from the
forward direction.

In the case of light, for which normally X « B, the shadow
scattering is not easily measurable since the shadow extends prac-
tically to infinity. In the nuclear case %/B is, say, y 6 or %, so
that the umbra or region of complete shadow extends only a short
distance back of the nucleus, certainly not as far back as the
measuring apparatus. Thus il is possible to make measurements
outside the main beam but still at small enough angles to it to
obtain the elastic shadow scattering. The existence and general
features of shadow scattering have been confirmed experimentally
by Kikuehi ot ah, Amaldi et ah, and Backer.

Charged Particles. The emission of charged particles such as
protons requires the penetration of a potential barrier. This
penetration probability is similar to that given in the theory of
a-decny and is quite small unless the emitted protons have energy
nearly equal to, or greater than, the barrier height B. Thus, in a
rough way, we may say that the protons must leave with a mini-
mum energy B. This would leave the residual nucleus at a lower
energy than if neutrons were emitted. Since the density of residual
nucleus states decreases rapidly with decreasing energy, the:
probability for proton emission will be much smaller than that for
neutron emission because of the fewer number of states available,
especially if the nuclear charge is high and the available energy low.

y-ray\$. The emission of 7-rays will in general be small com-
pared to heavy particle emission when the latter is energetically
possible because the coupling of the nucleus with the radiation
held involves the small factor e 2 /hc = 1/137,

DENSITY OF NUCLEAR ENERGY LEVELS-
NUCLEAR TEMPERATURE

The density of nuclear energy levels increases rapidly as a
function of energy. To see how this comes about a model which
is only a crude: approximation is used. We consider the nuclear

110

TOPICS NOT RELATED TO NUCLEAR FORCES

particles as independent of each other, and suppose each of them
has a set of equally spaced energy levels spaced by an energy
difference A. Then, the excited states of the .system will also be
spaced by the Interval A, and will have a greater statistical weight
the greater the excitation energy, because of the greater number
of ways of dividing the energy among the particles. When an
interaction among the particles is then introduced, there will be
splitting of each energy level ; and the statistical weight of an energy
level of the non-interacting system is a measure of the energy level
density in the same region of the spectrum, after the interaction
has been introduced.

To calculate the level density a model of the nucleus must be
used. Four models will be mentioned. (For more details set:
Nuclear Physics B, p. 79.)

1. Free Particles in a Box of the Size of the Nucleus. The level
spacing D is proportional to exp( — V E), where E is the excitation
energy of the nucleus. For A = 120, E = 3 Mev, we get D ~ 10
ev, which is about what is observed.

2. Free Particle in a Box, with Correlations. Bardeen has
pointed out that the free particle model must be modified to be
in accord with the assumption of exchange forces. The result
gives a level spaaing depending on excitation energy in about the
same way as before, but the level spacings are somewhat wdder:
D ~ 100 ev for A = 120, E = 8 Mev.

3. Lattice Model. This model is the opposite extreme of models
1 and 2, for the particles are here supposed to be firmly bound
and capable only of small vibrations about equilibrium. The re-
sults are similar to those for models 1 and 2. The level spacing is
proportional to expf-J^). For A = 120 and E = 8 Mev,
t>~ 100 ev.

4. Liquid Drop Model For heavy nuclei this model is quite a
good approximation. The level spacing is proportional to
exp(-i? V; ) for small S and exp(-/i H ) for larger E. For A = 120
and E = 8 Mev, D — 10 ev.

All these models give a level spacing which is a decreasing func-
tion of the energy of the form expL— /(£)], where f(E) is a slowly
variable function of the energy.

If the density of states, p{E) = 1/D, of any system is given
as a function of energy then an entropy can be defined as

THE COMPOUND NUCLEUS

117

£ = k log p(E), and a temperature as dS/dE = \/T(E). Each of
the four models mentioned will therefore define a nuclear tempera-
ture lis a function of excitation energy. It turns out that for
10 Mev excitation energy, hT is of the order of 1 Mev, i.e.,
T = I0 10o K.

The most satisfactory treatment of nuclear thermodynamics
(Weisskopf, Phys. Rev. 52, 295, 1937) avoids a model and supposes

I) = C exp(-L'Vi')

(213)

The constants B and C are determined from experiment: For low
excitation energies the exponential is close to 1 so that D is about
equal to C. From the observed position of the lowest excited levels,
it is found that:

For light nuclei (A ■
For heavy nuclei (A

20)
-200)

a-

io H

10 3

(214)

B can then be determined from neutron resonance levels near

E rv 8 Mev (binding energy of neutron in nucleus); this gives

B = 2 for light nuclei

(215)
B = 4 for heavy nuclei

if E is measured in Mev.

Any of the level density functions lead approximately to a
Boltzmann distribution for inelastically scattered neutrons. If
the incident energy of the neutrons is Bq and the energy of the
emitted neutrons is IF then tin.: excitation energy of the residual

nucleus is E n - W. Supposing that, the level density of the
residual nucleus is exp [+/(£)] and expanding,

m =/(e^ -rmw+ ■■■ (m

w : e get a level density

exp f(E) - exp f(E q ) X exp( -fW) (21 7)

Therefore, setting /' = 1/lcT (which is exactly the expression
demanded by dS/dE = l/T) gives a Boltzmann distribution for
the level density of the residual nucleus as a function of W and
therefore for the kinetic energies of the emitted neutrons. A more
careful consideration gives a probability of emission proportional

US TOPICS HOT RELATED TO NUCLEAR FORCES

to VW «K P (-W/kT) or W ^ 9 ^-W/kT) but experiment lias
not as yet given enough data to make it possible to distinguish
between them.

RESONANCE PHENOMENA
Let the energv levels of a nucleus Z A be as shown in Fig. 10 and
consider the process 7^ + i> - Z A U the incident neutron
lias exactly the right energy to form Z A in one or its excited states
the probability of capture is large. Such energies are called
resonance energies of the compound nucleus. The experimental

Vp- Resonances

_Binc!ing energy
""'of neutron

-^^.Ground state

Fig. 16. Energy levels
of a au&leus,

Pig. 17. Typical experimental

cross -seetiuii of a nucleus for

slow neutrons-

evidence (see Fig. 17) for neutron resonance energies m capture
processes led to the first theories of the compound nucleus .ex-
perimental ly, for A ~ 100, the level spacing D is about 10 ev, if
E is about the binding energy of the neutron, i.e., 8 Mev. D is
about the same at A ~ 200, and the appropriate binding energy
E ~ 5 Mev Tins can be understood because, on the one hand,
the number of particles is greater (and thus there are more possi-
bilities of distributing the energy) ; on the other hand the excita-
tion energy (binding energy of the particle) is smaller. For A
smaller than 100, the level spacing increases rapidly.
' There are several nuclei for which m0 re than one resonance is
knowm Among elements having only one (abundant) isotope,
In has 3 resolved resonances, I lias 5, and Ta 7. In addition,
of the experimental evidence was obtained by Rainwater, Havens,
and their collaborators, in several papers in Phys. Rev. 71 £1947).
In some; cases, onlv one resonance is observed; the level spacing
is then not directly known but it pan be taken as of the same order
of magnitude as the kinetic energy of the neutrons corresponding
to the first resonance.

THE COMPOUND NUCLEUS

11!)

For protons, capture resonances have been observed only for
the very light nuclei. The level spacings are of the order of 10
to 100 kev with an excitation energy of -10 Mev. lor heavier
nuclei the Coulomb barrier prevents capture resonances for
protons because the excitation energies which result after a proton
lias been given sufficient energy to get over the Coulomb barrier
are so high that the resonance levels overlap. A few resonances
have also been observed for a-particles, the reactions of which
lead mostly to the emission of protons or neutrons.

The width T of a nuclear energy level is defined as Y = n, r,
where r is the lifetime of the level. For most of the slow neutron
capture levels the width is about 0.1 ev. This can be decomposed

r=r 7 + r vl (218)

into the neutron width and the y-ray width. Almost all of r is
T which means that capture is far more probable than scattering
for slow neutron resonances.* T, t may be determined separately
in two different ways. First, the capture cross section at exact
resonance is given by

(const) X TjE r T

(219)

p is the width of the resonance at half-maximum; therefore, T\
can be determined from , at resonance, T and E T , Second the
ratio of scattering to capture cross sections at resonance is iyi T ,
and f, is very nearly equal to i\ Unfortunately, in order to get
the scattering cross section at resonance it must be ^entangled
from the potential scattering (Nuclear Physics B, p. 162) so that
this second method is ordinarily not ot much use. .

The first experiments on neutron capture were done by fiermi
and his collaborators, and by Moon and Tillman, using an ingen-
ious but rather complicated method: a neutron beam from which
impinged on an indium detector. Comparison of the radmactivi-
ties produced in this detector with and without an indium absorber
intervening, showed that neutrons which activated the indium
detector were strongly captured by the indium absorber It a
silver absorber was used instead, the absorption was small. On

* Mn has a Strong resonance at about 300 ev which gives mostly scattering
^"therefore has r n » iy TWs is lo be expected for light nuele: because
of their large level spacing; see equation 206a.-Ae(e added m prwf.

120

TOPICS NOT RELATED TO NUCLEAR FORCES

activity with and without the indium absorber, but with a silver
absorber the beam was very strongly attenuated. The conclusion
was that indium and silver were activated by neutrons of two
different energies. At present, the most satisfactory method con-
sists in using a modulated cyclotron beam and determining the
velocity of the neutrons by their time of flight to tin- target. For
very slow neutrons, a pile and a crystal spectrometer are often
preferable.

THE DISPERSION FORMULA

Breit and Wigner were the first to develop a theory of nuclear
resonance processes. The result was analogous to that in the
theory of optical dispersion

n - ■ 1 -j, (220)

The measurements using velocity selection can check the shape of
this curve and at the same time determine E T and P. To get the
coefficient of proportionality in equation 220, suppose that the
cross section cr is for the production of B with A incident. Then,
since the cross section is proportional to the half-width for disinte-
gration into B, it must contain fs, It also must contain T A for
symmetry reasons. This follows from the principle of detailed
balance: apart from statistical weights and a factor depending on
the ratios of momenta, o- A _> B should be equal to ff B -,A- (See
Chapter XL)

Finally we know that for the simplest case in which only one
kind of particle can be emitted or absorbed, T A = Fs = F, and
we know further that in this instance the largest possible cross
section for particles with I = is IttX 2 . Clearly, in the general
case, the wave length of the incident particle must occur. Collect-
ing all information,

a = %X, 2

Va Tu

(221)

(S - E r f + (l'/2) 3

This is known as the one-level Breit- Wigner formula. It gives
the correct dependence on momentum, in accord with the prin-
ciple of detailed balance

THE COMPOUND NUCLEUS

121.

For the dependence on the spin of the compound nucleus and
the generalization to more than one resonance level see Nuclear
Physics B, p. 101. There is only one instance in which the many-
level formula has been of use, namely,

He 4 + n -* He s -» He 4 + ft

(223)

which has two partly overlapping resonances near 1 Mev.

The dispersion formula has been derived many times. The
derivation must be quite different from, the treatment in optics,
where the interaction of the incident light and the atom can be
taken as a small perturbation.

For high-energy neutrons the dispersion theory goes over into
the statistical theory given previously. The partial widths of
the levels become of the order of magnitude of the level spacing
and the resonances are no longer observable.

For extremely slow neutrons, well below the first resonance,
Ta. is proportional to v (this follows from the fact that F A is propor-
tional to the density of states in momentum space, j/{dp/dE) rv -p)
and so the Breit-Wigner formula reduces to

X 2 v rv 1/s

(224)

This is the well-known .1 fv law for the cross section at very low
energy. It makes the number of processes per second, which is
av, independent of the energy distribution and proportional only
to the total particle: density. For very light nuclei, the spacing D
is very large and the 1/v law holds up considerable energies. For
gin + ?l ^B" it is valid to 50,000 ev. Absorption by B 10 is
therefore used for measuring neutron velocities.

APPENDIX: TABLE OF NUCLEAR SPECIES

Explanation" of the Table

Column 1:
Column 2:
Column 3:
Column 4:

Column 5:

K

e~

Z." Atomic number of the element.

Element." Chemical symbol of element.

A." Mass number of the Isotope.

Abund., per cent." Per cent abundance of isotope in the

afurally occurring element.

Disintegration." Symbols for nuclear processes are:

1 isomeric transition. (Emission of ->-rays or conversion
electrons.)
Electron capture.
Internal conversion electrons.
Negative, positive beta-particle emission.
a Alplia-particle emission.
n, H Emission of neutrons, protons.

V Denotes that the paitieular isotope has not been identi-
fied with complete certainty. Parentheses enclosing
one or more activities denote uncertainty in these, but
not in the identification of the isotope to which they
are assigned. Thus, 4; Ag 103 has been classified and found
definitely to have fi" activity; however, it is not certain
that 47 Ag 108 also has K-capture and conversion .elec-
trons. A comma setting off le" from one or more
symbols indicates that the conversion electrons belong
to the isomeric transition.
Columns and 7: Masses, with probable errors. A value in parentheses
indicates that the mass has been estimated from theory, the
isotope not having been produced as yet.
Column 8: Spin of the designated isotope.

Main Reffrenous

G. T, Scaborg, Table of Isotopes, Rev. Modern Phys. 16, I, 1944.

E. Segre, Isotope Chart, issued by Los Alamos Scientific Laboratory, 1 9*8,

In general, isotopes classified as A to D by Seaborg and Segre have been

included in this table, i.e., all those for which at least the assignment to a

definite element is certain.

123

121

Abund.,

APPEN
Disinte-

DIX

Error

Z Element,

A

per eeat

gration

Mass

X UJ 6

Spin

n

1

1.008 93

3

H

1 H

1

99.98

1.008 123

O.fi

H

2

. 02

2.014 70S

1.1

i

3

(T

3.017 02

3.4

H

2 He

a

~10~ 5

3.017 00

4

4

100

4.003 90

3

5

n

5.013 7

35

r

6.020 9

50

3 Li

5

ii

(5.013 6)

60

6

7.5

0.016 97

5

l

7

92 . 5

7.018 22

6

n

8

f

8.025 02

7

4 Br

()

(6.021 9)

100

7

K

7.019 16

7

8

tx

8.007 85

7

9

100

9.015 03

6

%

10

p-

10.010 77

8

11

(11.027 7)

5 B

9

9.016 20

7

10

18.4

lO.Olfi 18

9

i

11

81.6

11 .012 84

8

H

12

r

12.019

70

13

(13.020 7}

6 C

10

(? +

10.021

30

11

[i+

11.014 95

9

12

98.9

12.003 82

4

IS

1.1

13.007 51

K)

H

14

p-

14.007 67

5

15

(15.016 5)

7 N"

12

(12.023 3)

13

+

13. 009 88

7

U

99 . 62

14.007 51

4

i

15

0.3S

15,004 89

21

H

Hi

r

> 16. 006 5
< 16.011

17

(17.014)

8

14

(14.013 1}

15

v +

15.007 8

40

IS

99.757

16.000 000

Standard

17

0.039

17.004 50

a

IS

0,204

IS. 004 9

40

19

P~

19.013 9

TABLE OF NUCLEAR 3PECIE8

Abund.,

Disinte-

Error

z

Element

A

per cent

gration

Mass

X 10*

9

F

16

(16.017 5)

17

ff*

17.007 S

30

18

.3+

18.006 5

00

19

100

19.004 50

26

20

r

> 20. 004 2
<20.009 2

21

(21.005 9)

10

No

18

(18.011 4)

19

+

19.007 81.

20

20

90.00

19.998 77

10

21

0.27

20.999 03

22

22

9.73

21.998 44

30

23

r

23.001 3

11

Na

21

p +

21.003 5

22

\$+

21.999 9

50

23

100

22.996 18

31

24

a~

23.997 5

45

25

Ufr

(24.996 7)

12

Mg

22

(22.006 2)

23

ft +

23.000 2

40

24

77.4

23.992 5

00

25

11.5

24.993 8

90

26

11.1

25.9S9 8

50

27

r

26.992 8

150

13

Al

25

3 +

24.998 1

100

20

P +

25.992 9

150

27

100

26.989 9

80

28

&~

27.990 3

70

29

P~

28.989 3

80

30

(29.995 4)

1.4

Si

27

P +

26.994 9

90

28

89.6

27.986 8

60

29

6.2

28.986 6

70

30

4.2

29.983 2

90

31

P~

30.986 2

(iO

32

(31.964 9)

15

P

29

e+

28.991 9

100

30

p +

29.9S7 3

10

31

100

30.984 3

50

32

r

31.9S2 7

40

33

(32.982 6)

34

\$-

33.9S2 6

40

125

Spir

H

;

126

APPENDIX

A bund.,

DltfMLilV

Error

Z

Element

A

per cent

gration

Mass

X 10 s

Spin

m

S

31

p +

30.989 9

32

95.1

31.

980 89

7

33

0.74

32.

980

GO

34

4.2

33.

977 10

33

35

F

31.

978 8

SO

38

0.016

35.

97S

100

37

§~

36.

982 1

30

17

CI

33

#

32

986

34

P

33

980 1

200

35

75.4

34

978 67

21

H

36

<3 + ,S" K

35

97S 8

100

37

24.0

36

977 50

11

%

38

fr

37

981

300

39

(3S

979 4.)

18

A

35

r

3-1

985

36

0.307

35

978

100

37

K

36

977 7

38

0.061

37

974

250

3D

(38

975 5}

4!)

99.632

39

975 6

60

41

r

10

977

60

19

K

37

(36.983 0)

38

3 +

37

.979 5

39

93.38

38

.974 7

%

40

0.01.2

e-x

39

.976

100

l

41

6.61

10

.974

%

42

w

■13

Ufr

20

Ca

39

VP*

40

96.96

39

.975 3

150

(.)

41

TJ K g-

42

0.64

41

.971 1

43

0.15

■12

.972 3

■11

2.06

■15

r

46

0.0033

48

0.19

49

r

21

Bo

41
43
It

45

100

44

.966 9

so

%

46

S~ if

Z Element

21 Sc
{cant.)

22 Ti

23 V

24 Cr

25 Mn

26 Ee

27 Co

TABLE OF NUCLEAR SPECIES

A
47
48
49

15
46
47
48
49
50
51

47
48
49
50
51
52

49
50

51
52
S3
54

55

51

52
54
55
56

53

54
55
56
57
58
59

55
56
57
58
59
60

Abund,,
per cent

7.95

7.75

73.45

5.51

5.34

100

4.49

83.78
9.43

2.30

100

0.04

91.57
2.11
0.28

Disinte-
gration

r

(3+

U ,3 '"
£+ K

a k

r

U K cT

V

f3-'~K
K

100

a+ k
\$+ K r
13+ K

Mass

45.966 1
46.964 7

47.963 1

48.964 6
49.952 1
50.958 7

50.957 7

50.958
51.956

52.956

54.957

53.957

Error
X 10 5

127

Spin

100

100

50

60

40

100

50

%

55.956 8 170

56 . 957

n

128

ArPENDIX

Abund.,

Disinte-

Error

'/, EI c men t,

A

per cent

gration

Mass

X 10 &

28 M

57

&

58

67.1

57.959 4

10

59

V .a '

GO

26.7

59.949 S

40

61

1.2

60,953 7

150

62

3,g

61.949 3

40

ea

r

0-1

0.88

63.947 1.

60

29 Cu

58
BO
01
62

e+

0+ K

P +

63

70.1,3

62.957

400

G4

r& K

56

29.87

64.955

400

66

r

30 Zn

63

^

64

50 . '.)

63.955

400

65

fl • K f

m

27.3

65.954

400

67

3.9

66.954

400

68

17.4

67.955

300

69

li~ I

70

. 5

69.954

300

31 Ga

(it
65

m

67
68

U 3+
K '~e~
fi +
K ,.r

69

61.2

68 . 952

800

70

8~

71

38.8

70.952

900

72

0~

71

V fi-

12 Ge

69

70
71
72
73

74
75
76
77
78

21.2

27.3

7.9

37.1

fi.5

ll

3+ A* ( -

«r

Spin

H

%

%

TABLE OF NUCLEAR SPECIES

Mass

AbuTiJ.,

Disinte-

z

Element

A

per cent

gration

33

As

72

73
74
75
76
77
78

100

U 3+

V P + K e

r ? +

&- /3 + K
Up~

34

He

74

75
76
77
78
79
80
82
83

0.9

9.5

8.3

2-1.0

48.0
9.3

K e"

35

Br

78

79
80
SI
82
83
84
85
87

50-6
49.4

3~, 1 <T

r

r
r

129

Error

X 10 5 Spin

%

M

m

Kr 78
79
80

81
82

0.35
2 01

1 I . 53

(./ 1 <~

83

] 1 . 53

i «r

%

84

57.11

85

p~

86

17.47

87

u tr

88

0-

89

,a-

90

U fi-

*

91

ll fr

92

ur

94

U fi-

95

ll 07

^EJ '

J^™

130

Abund.,

APPENDIX

Disinte-

Z

Element

A

per cent

gration

37

Rb

82
84
85
86

72.8

V
U

0~

87

27.2

P~

88

tr

gg

r

'JO

V fi-

91

ll rr

92

u&-

!)4

U ii~

95

v\$r

38

gt

84
85
8(5

0.50
9.86

J <r K

87

7.02

le-

88

82.50

89

ft"

90

UfT

01

U fi-

92

ll fi-

01

ll ft-

95

ur

39

Y

86

87
88
89
90
91
92
04
95

100

UK
(l e~) K

V fi~, t e-
U fi-
ll fi-
ll r

40

Zr

89
90
91
92

03
94
95

48.0
11.5

22.0

17.0

fi~, I or B
U0-

v a~

■11

Cb

96

07

90

91
92

Mass

Error
X 10*

Spin
%

TABLE OF NUCLEAR SPECIES

ur

U8+

UK.tr

W

A bund.,

Disinte-

Error

Z Element

A

per tent

gration

Mass

X 10 5

41 Cb

93

100

(sent.)

94

95
96

97

a

U 3~, I cr

U

42 Mo

92

03

14.9

U (3+

94

9,4

93.945

800

95

16.1

94.946

800

00

J6.G

95.944

800

97

9 . 05

96 . 945

900

08

24.1

97.943

000

09

v r

100

9.25

101

u r

102

U fi-

43 Te

96

98

99

101

102

ll K

U K <T
6", I e~
U fi-
ll r

41 Ru

90

5.08

95.945

1100

08

2.22

97.943

1100

00

12.81

98 . 944

1100

100

12.70

99.942

1100

101

16.98

100.946

1100

102

31.34

101.941

1100

103

r

104

18.27

105

fT

106

u r

107

O \$r

45 Rh

102

s-ft +

103

100

I

102.941

1100

104

li~, lf~

105

\$r

106

U fir

107

rye-

46 Pd

102

0.8

101.941

1100

104

0.3

103.941

1100

105

22.6

104.942

900

106

27.2

105.941

1000

108

26.8

107.941

1000

131

Spin

%

H

132

APPEXDIX

Abund

., Ksiiite-

Error

Z

Element A

per ecu

t grat

ion

Mnss

X 10 s

Spin

46

Pd

109

u tr

(cont.)

110
111
112

13,5

pr

109.941

1000

■17

Ag

105
106

U K

3+ K B~

107

SI. 'J

{/O

106.945

600

] A

108

r (K E

")

109

48.1

108.944

700

k

no

8~

in

P~

112

w~

m

Cd

106
107

108
110
111
112
113
114
IIS
116
117
1.18

1.4

1.0

12. S
13.0
24.2
12.3

2S.0

7.3

UK
U, I ,-

49

In

no
.1 11

112
113

114
1.15

no

11.7

4.5
95.5

U 8+

V a+ «-

U, I e~,
I er

1 e~, 3~

1 <r
a~

Ke~

50

Sn

112
11.3

114

1.1

O.S

KtT

115

0.4

114.940

1400

H

116

15.5

115.939

1400

11.7

9.1

ne. 937

1400

H

US

22.5

117.937

1400

110

9.8

118.938

1400

H

120

28.5

119.937

1400

121

U-ftr

122

5.5

121.945

1400

123

U8-

TABLE OF NUCLEAR SPECIES

Abund.,

Disinl.c:-

Error

z

Element

A

per

cent

gration

Mass X 10 5

50

Si)

124

6

8

123.944 1400

{cmd.)

125
127
128

if fir

U 3~
11 3-

51

m

120
121
122
123
124
126
127
128
129
132
133
136

56

44

ti-
ll fir

&~
&~

U3~

u 3-

52

Te

120
121

Q

088

IK<T)

122

2

43

(ier)

123

85

124

4

59

125

6

93

126

18

71

127

0~, 1 e~

128

31

86

129

8~, I <r

130

34

52

131

8~, I e~

132

tiff"

133

u 3-

135

fi~

tae

u tr

137

,

u a~

53

1

124
1 26
127
128
130
131
132
133
135
136
137

100

3+

a~ K

a-
a-
3-
Vb-

V 8-

u 3-

133

Spin

%

J

4

APP1

Abund.,

2NDIX

Disinte-

Error

z

Element

A

per

cent

gration Mass X 10 s Bpin

54

Xe

124
126
127

128

1

094
088

90

U, I e~

129

26

23

u

130

4.07

131

21

17

H

132

26

96

133

&)lr

134

10

54

135

(Tl

136

8

95

137

U P~

138

Up~

139

sr

140

r

141

0-

143

u tr

144

Up~

55

Qa

130
132

133
134
136
137
138
139
140
141
142
143

100

V

v k <r

Uff-

u p-

pr

Up-

P~

v tr

u

%

50

Ba

130
132
133
134
135

•2
8

101
097

42
59

I er

%

136

7

81

137

11

32

%

138

71

.66

G

139

p~

140

p~

141

p-

142

ur

143

ur

145

up-

TABLE OF NUCLEAR SPECIES

135

Abund.,

Disinte-

Error

z

Element

A

per cent

gration

Mass

X 10 5

Spin

57

L:i

137

U K

139

100

138.953

800

K

140

r

141

P~

143

U 3~

144

up-

145

Up~

58

Ce

136
138
140
141
142
143
L44
145
147

<1

<1
89

11

r

Up~

u p-
u p-

U p~

59

Pr

140
[41
142
143
144
145
147

100

p +

p~

u r

up-

Up~
UfC

M

60

Nd

141

142
143
144

25.95

13.0

22.6

p +

145

9.2

144.962

400

146

16.5

145.962

400

148

6.8

147.962

400

150

5.95

149.964

400

61

61

143
144

145
146

147

Up~
U I or K

Up~

u
up-

62

Sm

144
146
147

1.48
149

3

16.1
14. 2
15.5

VI

a

136

A

PPEXDIX

Abund.,

Disinte-

2

Elemenl

A

per cent

gration

62

Sm

150

11.6

(cont.)

151

u-fir

152

20.7

154

18.!.)

S3

Eh

151
152
153
154
155
156
157
158

49.1
50.9

U fi~ e~

Utf

V ti-
ll ti-
ll \$-

64

Gd

152

154
155
156
157
158
160

0.2
1.5
18.4
19.9
18,9
20.9
20.2

66

Tb

159
160

100

fr

OG

Dy

158
160
161
162
163
164
165

XI. 1

0.1
21.1
26.6
24.8
27.3

it

67

Ho

165
166

100

u &-

88

Er

162

164
166
167

168
166
170

0.1

1,5

32.9

24.4

26.9

14.2

USi~

6ft

Tm

169

170

100

03-)

Mass

Error

X IB 6

Spin

m

TABLE OF XUCLEAR SPECIES

Mass

153.971

600

154.971

600

155.972

600

156.973

600

157.073

600

159.971

600

%

H

Abund.,

Disinte-

z

Element

A

per cent

gration

70

Yb

168
170
17!
172
173
174
175
176

0.08

4.21
14.26
21.49
17.02

29.58

13.38

U

71

Lu

175

97.5

176

2.5

0- K

177

if ft

72

Hf

174
176
177
178
179
180
181

0.18
5.30
18.17
27.10
13.85
35.11

li~

73

Til

180

(.3") IS. e

181

100

I

182

ft

74

W

180
182
183

184
185
186

187

—0.2
22 . 6

17.3
30.1

29.8

u p-
w'fr

75

Re

184
185
186
1S7

188

38.2
61.8

UK
U P~

u p-

76

Os

184
186

0.018

1.59

187

1.64

K

188

13.3

189

16 . 1

190

26.4

191

v p-

192

41.0

193

Ujr

Error
X 10 s

137

Spin

X

>7

<%

<H

7 A

3k

%

189.04 2000 Viar%

190.03 2000

192.04 2000

13S

AI
Abund.,

PENDIX

Ilisinlr-

Error

Z

Element

A

per cent

graf.ion

Mass

X 10 s

Spin

77

Ir

191
192

3S.5

r

191 .04

2000

H

193

61. 5

193.04

2000

%

194

P~

78

Pt

192

0.8

194

30.2

194.039

1400

195

35.3

195.039

1400

M

198

26.6

(/O

196.039

1400

197

jjf

198

7.2

198.05

2000

199

r

79

An

196

TJ \$T e~

197

100

I

197.04

1000

%

198

§r e~

199

l3~

200

v&-

SO

Sg

190
197
198

0.15

10.1

Ke~

199

17.0

I er

H

200

23.3

201

13.2

U

202

29.6

203

ff'tr

204

0.7

205

ir

81

Tl

198

199
202

U K f
V K "e~
U K t~

203

29.1

203-05

2000

H

204

Upr

205

70.9

205 . 05

2000

%

206

ur

AcC"

207

r

ThC

208

r

Tl

209

p-

RaC

210

e-

82

Pfa

203

u \$+

204

1.5

204.05

2000

205

v, i r

206

23.6

206.05

2000

207

22.6

207.05

2000

}£

TABLE OF NUCLEAR SPECIES

Abi

ind., 11 i si ii fce-

Error

z

Element

A per

eent gration

Mass

X to?

82

Pb

208 52.3

208.05

2000

icont.)

209

8~

RaJD

210

tt

AcB

21]

0-

ThB

212

g"

Pb

213

fr

SaB

214

r

S3

Bi

207

K sr

209 100

209.05

2000

RaE

210

i'~

AcC

211

ft" a

ThC

212

§T\M,

Bi

213

p~ a

Rati

214

f}" a

84

Po

210

a

AcC

211

a

ThC

212

a

Po

213

a

RaC

214

a

AoA

215

<x

ThA

210

(i~a

Po

217

a

RaA

218

r a

85

At

21 I

K a

88

An

219

&

Tn

220

a

Rn

221

a

Rn

222

a

87

87(AeK)

22;;

v pr

88

AcX

223

a

ThX

224

a.

Ra

225

a

Ra

226

a

MsTbi

228

v~

m

Ac

227

fi- <z

MsTh 2

228

l3~ a

90

RdAo

227

a

RdTh

228

a

Th

229

a

To

230

a

UY

231

\$r

139

:pm

140

APPENDIX

Ab unci.,

Disinte-

Error

Z Element

A

per cent

gration

Mass

X io 5

Spin

90 Th

232

100

a

232.11

3000

(cord.) Th

233

r

UXi

234

&"

91 Pa

231
232
233

a

{j- -

X

m

234

r

VX-i

234

wi

92 U

233

a

UII

234

0.00518

a

AcU

235

0.719

a

u

237

r

UI

238

99.274

a

238.12

3000

u

23'.)

r

93 Np

234
235
256
237
238
239

K
K

r

ix
ST

r

94 Pii

238
239

a

a

95 Am

241

a

96 Cm

240

242

®

^

INDEX

Allen, 22

Allowed transitions ( h <?-decay), 97
in intermediately heavy nuclei, 104
in mirror nuclei, 103
a-parlicle, rote of, 84
m-parlicle emission, 14
Alternation, intensity, in band spec-
tra, 18
Alvarez, 24, 53, 54
Amaldi, 91, 93, 115
Angular distribution, of neutron-
proton scattering, 39
at high energy, 03
of proton-proton scattering, 69
Angular momentum, 38
Annihilation, of electrons and posi-
trons, 2
of heavy particles, 2
Arnold, 25, 28
Atomic weight, 3
Atoms, theory of, 23
Axial vector, 107

Backer, 115
Backus, 102
Bailey, 44
Band spectra, 15
Barrier, potential, 7

penetration of, 115
Bartlett force, 82
Beams, molecular, 15
Bennett, 44
Benjstralh, 44
Be 8 , 5
^disintegration, 10, 97

and neutrino, 20

Fermi theory of, 97

^-disintegration, Konopiriski-U hi en-
beck theory of, 100

Kurie plot of, 100

of neutron, 105

selection rules in, 99
/3-spectrum, 20
Bcihc, 105
Binding energy, of deuteron, 29

of neutron in heavy nuclei, 1 1 7
Binding of proton in molecule, effect

of, 47
Block, 15,24
Bohm., 44
Born, 37

Bom's approximation, 18, 109
Bose statistics, 16
Breit, 69, 70, 71, 120
Breit-Wigrier formula, 1 20
Bretscher, 60
Brickwe.dde, 53
Britlouirt, 7
Erode, 94
Bulk matter, internal energies of, SO

Capture of neutrons, by protons, 00,
79
in heavy nuclei, 1 18
Cenler-of-mass coordinates, 37
Champion, 92

Charge, I

Charged meson theory, 88, 95

Charged particles, nuclear reactions

involving, 8
Chemical bond effect on neutron

scattering, 47
Chemical properties, 1
Compound nucleus, 109
disintegration of, 1 12

141

INDEX

143

142

INDEX

'

fl I

1 1!
Ill

,
ill

Compound n&ofcja, formation of, 111

Compton wave length, 94
Condon,

Conservatism of energy, 5
Constituents of nuclei, 8
Cork, 100
Coulomb field, 101
Coulomb repulsion, 6, 8
Coulomb scattering, 64, 65
Critchjidd, 105

Cross section, for nuclear reactions,
8, 114
for capture of neutrons by protons,

B0, 79
for photoelectric disintegration of

deuteron, 56, 79
for scattering, of neutrons by heavy
nuclei, 7, 114
of neutrons by protons, 37, 40, 79
of protons by protons, 64
geometrical, 7

for capture by heavy nuclei, 111
total, 40

D-D neutrons, 42
de Brogiie wave length, 8
of electrons, 9
of neutron or proton, 8
do Brogiie wave number, 37
Decay, £-, see ^-disintegration
Decay processes, 13
Dee, 40

Density of nuclear energy levels, 115
Depth of nuclear potential well, 32, 70
Determination of force constants, 70
Deuteron, excited slates of, 34

ground state of, 29

magnetic moment of, 26

photo disintegration of, 79

physical properties of, 25

singlet state of, 43

states of, with tensor forces, 74

virtual state of, 45

wave function of, 33
DeWirc, 53

Dirac S-function, 95
Disintegration, p, 10, 97

nuclear, 13

of compound nucleus, 112
D i sin I .e gra li on probabi 1 i I i es , 1 1 3
Dispersion formula, 120
Dispersion theory, 121
Distribution, angular, at high energy,

63
Dunning, 53

Einstein's relation, 2

Eisenbud, 69, 70, 87

Electric dipole moments in nuclei,

absence of, 75
Electron energies, distribution of, in

^-disintegration, 18
Electrons, annihilation of, 2
in nucleus, 5, 9
spontaneous emission of, 1.0
Electrostatic interaction of protons, 7
Emission, of a-particles, 14
of /3-rays, 10, 14
of 7-rays, 13
of heavy particles, 13
of light quanta, 14
of neutrons, 13
of protons, 14
Energy, 5
conserved, 5
kinetic, 5
Energy distribution in inelastic scat-
tering, 114
Energy equivalent, 2
Energy levels, nuclear, density of, 115
Exchange and spin, relation between,

84
Exchange forces, 81
and saturation, 83
effects of, 82

quantitative theory of, 87
Excited states, of deuteron, 34

of nucleus, 111
Experiments, on neutron-proton scat-
tering, 42, 91
on photo disintegration, 60
on scattering by para-hydrogen, 49

tages of, 64

East neutrons, 7

Fast nuclear particles, uncertainty in

position of, 111
Feather, 60
Fermi, 43

Eermi interaction, 97
Fermi statistics, 16
Fisk, 81

Forbidden transitions, half-lives, 106
Force, Bartlett, 82
Heisenberg, 82
Majorana, 82
Wigner, 82
Force constants, determination of, 76
Forces, exchange, 81

quantitative theory of, 87
non-central, 73
nuclear, meson theory of, 94
ordinary, B on-sat urati on of, 81
prol.on-prol'.on, 64
saturation of, 80
short-range, 66
Formula, dispersion, 1 20
Free particle model, 116
Frixch, 44
Frostier, 94
Fundamental particles in nucleus, G

7-rays, emission of, 13

Gamow, 6, 106

Gamow-Teller selection rules, 106

Geometrioal cross section, 7, 111

Gurlach, 15

Gilbert, 40

Go'ppert^Mayer, 33

Goldhaber, 29, 60

Gordon, 94

Graham, 60, 111

Ground state, of deuteron, 29

of He 2 , (54
Gurney, 6
Gyromagnetic ratio, 24

Halban, 60

Ilamermesh, 53

Hansen, 15

H ovens, 118

Haxby, 20

Heist'ttberg, 81

Heisenberg force, 82

H eider, 56

He-\ 5

Herh, 70

Heydenburg, 66

High-energy neutrons, experiments

with, 91
High-energy phenomena, information

from, 62
Hyperfine structure of spectra, 15

Inelastic scattering, energy distribu-
tion in, 114
Information obtainable from high-
energy experiment, 63
Intensity alternation in band spectra,

18
Interaction, electrostatic, of protons, 7

Interference between nuclear and

Coulomb scattering, 64
Invariance, against inversion, 73

relalivistie, 107
Inversion of coordinate system, 73
Isobars, 2, 1 1

of neighboring Z, 1.1
stability of , 1 1
"Isomer of nucleus, 13
Isotopes, defined, 1
Isotopic spin, 84
Isotopic spin functions, 85

Jentschke, 29

K electron capture
Kawne, 71
Kellogg, 23
Kemhle, 74
Keener, 78
Kerst, 70
Kikuchi, 101
Kimura, 29
Kinetic energy, 5

10, 14, 22, 108

115

mr

144

INDEX

Klei a -Gordon equal, ion, 94
Kmuypinski, 97, 103

Konopinsld-Uhlenbeek theory, 100,

104
Kramers, 7
Krugcr, 92
Kurie plot, 100

Lande's formula, 28

Lmmhlin, 92

Lawson, 100

Levels, n u clear energy, density of, 115
width, experimental, 119

in allowed transitions, 102
in forbidden transitions, 105

LigM quanta, emission of, 14, 119

Liquid-drop model, 80

Li 5 , 5

Low-energy phenomena, information

from, 62
Lyman, 101

Magnetic moments, 15

of deuteroii, 25
Majorana forte, 82
iliti.j7i.fle/er, 29
Mass, reduced, 37
M.ass excess, 4
Mass number, I

Mass spectrograph, 3
Mosses, 37, 66, 67

Matter, bulk, internal energies of, 80

Meson theory of nuclear forces, 88, 94

Metastable state of nucleus, 13

Molecular beams, 15

Molecular velocity effect on neutron
scattering, 49

Moment, magnetic, IS

Moon, 1 19

Morse, 81

Mott, 37, 66, 67

Myers, 29

Neutral meson theory, 88
Neutrino, 20

Neutrino, /3-disintcg ration and, 20

experimental evidence for, 21
Neutrino mass, 20
Neutron, p-decay of, 105.
physical properties of, 24
slow, cross section of, 43
wave length of, 7
Neutron emission, 13
Neutron-proton scattering, 78
experimental results on, 42, 69
at high energies, 91
Neutron spin, 45
evidence for, 45
from scattering, 45
Neutrons, capture of, by protons, 60,
79
Li-D, 42
fast, 7

in nucleus, 5, 8

scattering of, by ortho-hydrogen,
49
by para-hydrogen, 49
by protons, 37, 78

at high energies, 91
by protons bound in molecules,
47
thermal, 43
Non-central forces, 73
Non-saturation of ordinary forces,

81
Non-zero spin, nuclei of, 18
Nuclear abundance, 12
Nuclear charge, 1
Nuclear constituents, 15
Nuclear disintegration, 13
Nuclear energy levels, density of, 1 15

width, 119
Nuclear force, between neutron and
neutron, IS, 71
between neutron and proton, 30
between proton and proton, 64, 71
meson theory of, 94
saturation of, 80
spin dependence of, 45
Nuclear reactions, cross section for,
114
general theory of, 110, 114
involving charged particles, 8, 115

INDEX

145

Nuclear scattering, 7, 64, 67, 114
interference between Coulomb scat-
tering and, 64
Nuclear species, table of, 123
Nuclear spin, 15, 23
Nuclear temperature, 115
Nucleus, absence of electric dipole
moments in, 75
basic facts on, 1
oompouTirl, L09

disintegration of, H2
formation of, 111
constituents of, S
excited states of, lit
fundamental particles in, 5
isomer of, 13

magnetic moment of, 15, 25
metas table state of , 13
residual, 111
size of,

stable, regularities in, 2
Nuckolls, 44
Number, mass, 1

Grtho-hydrogen, scattering of neu-
trons by, 49

Packing fraction, 4

Para-hydrogen, scattering of neu-
trons by, 49

Parity of wave function, 75

Parkinson, 70

Partial width of level, 1 1 4

Particles, charged, nuclear reactions
involving, 8
heavy, annihilation of, 2
emission of, 13

Pauli principle with isotopic spin, 86

Peierls, 78

Penetration of potential barrier, 115

Phase slrifts, 37

for I ?s 0, 38
Photodisintegration, 56, 79

experiments on, 60
Photoelectric effect, 56

Photo magnetic effect, 57
Physical properties, of deuteron, 25
of neutron, 24
of proton, 23
Titter, 53, 54
Plain, 70

Polar four vector, 107
Position of fast- nuclear particles, un-
certainty in, 11 1
Positrons, annihilation of, 2

spontaneous emission of, 10
Potential, relation between range and

depth of, 32, 70
Potential barrier, penetration of,

115
Potential well, rectangular, between
two protons, 70
of deuteron, 31
Powell, 92

Probability, disintegration, 113
penetration, 115
sticking, 111
Proof of saturation, 84
Properties, chemical, 1
Proton, physical properties o) , 23
Proton emission, 14
tages of, 64
Proton-proton forces, 64
Proton-proton scattering, 64
experiments on, 70
theory of, 65
Proton resonances, 119
Protons, bound in molecules, scatter-
ing of neutrons by, 47
capture of neutrons by, 60, 79
in nucleus, 5, 8
scattering of, by protons, 64
scattering of neutrons by, 37
Pseu do scalar, 107
Purcell, 15

1

Quantitative theory, of exchange
forces, 87
of nuclear forces, 23
Quantum mechanics, 23

14S

INDEX

Rabi, 15, 23, 27

interaction of deuteron with, 56
ootopole, 13

if, 10, 97
Ragan, 71
Rainwater, 1 IS
Ramsey, 23
Range, of nuclear forces, 32, 49, 70

ia relation to depth, 32
Rari.ta, 27, 59, SO, 73, 76, 77, 87, 88,

89
Reactions, nuclear, involving charged

particles, 8
Recoil energy, nuclear, 21
Rectangular potential well, 31, 70
Reduced mass, 37
Regularities in stable nuelei, 2
Relation, between range and depth of
potential, 32
between spin and exchange, 84
Relativistic in variance, 107
Residual nuclei, 111
Resonance phenomena, 118
Resonances, proton, 119
Richards, 44
R:idiardson, 101
Rich-man, 44
Roberts, 26, 28
Role of ^-particle, 84
Rotation of coordinate system, 73
Rules, selection, Fermi, 99

Gamow-Teller, 106
Rutherford, 65

Sachs, 33

Saturation, of nuclear forces, 80

proof of, 84
Seaixrrg, 111, 123
Segre, 123

Scalar in /5-theory, 107
Scattering, Coulomb, fl'l

interference between nuclear
and, 01

inelastic, energy distribution in, 1 1 4

nuclear', 64, 114

Scattering, of neutrons, by heavy nu-
clei, 7, 1 14
by ortho-hydrogen, 49
by para-hydrogen, 49
by protons, 37, 78

angular distribution of, at

high energy, 63
- experimental results on, 43, 91
spherical symmetry of, 39
total cross section for, 40
by protons bound in molecules,
47
of protons by protons, 64

angular distribution of, 69

spherical symmetry of, 39
Scherrer, 102, 108
Schdff, 81

Schrodinger equation, 31
Schwinger, 27, 28, 49, 50, 53, 59, 60,

73, 76, 77, 87, 88, 89
Selection rules, Fermi, 99
in /3-dccay, 99
Gamow-Teller, 106
Short range of nuclear force, 30, 60
Skoupp, 20
Singlet state, of deuteron, 43

sign of energy in, 45
Size of nucleus, 6
Slow neutron cross section, 43
Spectra, band, 15
Spectral lines, splitting of, 15
Spin, 15

and exchange, relation

between, 84
and isotopic spin, 84
and statistics, 15
iso topic, 86
nuclear, 15
Spin dependence of nuclear force, 45
Splitting of spectral hues, 15
Spontaneousemission,offf-particles,6
of electrons, 10
of positrons, 10
Spherical symmetry of scattering, 39
Stability, of isobars, 11
of nuclei, 5

INDEX

147

Stable nuclei, regularities in, 2
States, excited, of nucleus, 111

of deuteron, 74
Statistical considerations, 60
Statistical weights, 01
Statistics, 10

Bose, 16

Fermi, 16

of neutrons, 1 S

of protons, 18

spin and, 15
Staub, 40

Stern-Gcrlach experiment, 15
Stetier, 29
Stevens, 20, 30
Sticking probability, 111.
Structure of spectra, hypcrfine, 15
Survey of low-energy phenomena of

deuteron, 62
Sutton, 53

Symmetric meson theory, 88
Symmetry of wave funciton, 68

Tashelc, 71
Teller, 49, 50, 106
Temperature, nuclear, 1 15
Tensor in /3-theory, 107
Tensor forces, 28, 75
Thaxton, 69, 70
Theory, of atoms, 23

of exchange forces, 87
Thermal neutrons, 43
Thomas, 31
Tillman, 1 19
Total cross section for scattering of

neutrons by protons, 40
Townsend, 101

Transitions, allowed, in /3-d i si nteg ra-
ti on, 99

7-ra.y, 13

Transmission coefficient of barrier, 7
Time, 66
Tyler, 100

UMeribeck, 97

Uncertainty in position, 111

l/v law, 121

Van Attn, 29

Van de Graaff machine, 29

Variation principle, 80

Virtual state of deuteron, 15

VolkajJ, 82

Wave function, of deuteron, 33
parity of, 75
symmetry of, 68
Wave functions, relativistie eom-

bi nations of, 107
Wave length of neutron, 7
Wave number, de Broglie, 37
Wei glits, statistical, 61
Wei.sskopf, 113, 117
Wells, 20
Wentz el -B rilloui n-Kramers method,

7
White, 66
Wick, 93
Width of level, 113

partial, 1 14
Wiendenbeck, 29
Wianer, 28, 30, 43, 44, 45, 52, 73, S4,

87, 120

forces, 30, 66
Wigner forces, 81
Williams, 44
Wilson, 70, 92

Yukawa, 94

Zaeharias, 23

*

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